Toponogov's Theorem and ApplicationsbyWolfgang Meyer

These notes have been prepared for a series of lectures given at the College onDi�erential Geometry at Trieste in the Fall of 1989. The lectures center around To-ponogov's triangle comparison theorem, critical point theory and applications. In theshort amount of time available not all the aspects can be covered. We focus on thoseapplications which seem to be most important and at the same time most suitablefor an exposition. Some basic knowledge in geometry will be assumed. It has beenprovided by K. Grove in the �rst series of these lectures. Nevertheless we try to keepthe lectures selfcontained and independent as much as possible. For the result aboutthe sum of Betti numbers in section 3.5 a lemma from algebraic topology is needed. Aproof for this result has been provided in the appendix.I am indebted to U. Abresch for many helpful conversations and also for writingand typing the appendix. 1

Contents1 Review of notation and some tools 21.1 Covariant derivatives : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21.2 Jacobi �elds : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41.3 Interpretation of curvature in terms of the distance function : : : : : : 51.4 The levels of a distance function : : : : : : : : : : : : : : : : : : : : : : 91.5 Data in the constant curvature model spaces : : : : : : : : : : : : : : : 101.6 The Riccati comparison argument : : : : : : : : : : : : : : : : : : : : : 122 The Toponogov Theorem 143 Applications of Toponogov's Theorem 213.1 An estimate for the number of generators for the fundametal group : : 213.2 Critical points of distance functions : : : : : : : : : : : : : : : : : : : : 233.3 The diameter sphere theorem : : : : : : : : : : : : : : : : : : : : : : : 283.4 A critical point lemma and a �niteness result : : : : : : : : : : : : : : : 303.5 An estimate for the sum of Betti numbers : : : : : : : : : : : : : : : : 333.6 The soul theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 394 Appendix: A topological Lemma 471 Review of notation and some tools1.1 Covariant derivativesWe consider a complete Riemannian manifold M with tangent bundle TM and Rie-mannian metric h ; i and corresponding covariant derivative r of Levi Civita, whichis the unique torsion free connection for which h ; i is parallel, i.e. for any vector �eldsX; Y; Z on M we have rXY � rYX = [X; Y ] (1)and X hY; Zi = hrXY ; Zi+ hY; rXZ i : (2)The last two equations are equivalent to the Levi Civita equation2 hrXY ; Zi = X hY; Zi+ Y hZ;Xi � Z hX; Y i2

+ hZ; [X; Y ]i+ hY; [Z;X]i � hX; [Y; Z]i (3)If ~M is an arbitrary manifold and f : ~M ! M a di�erentiable map, f� : T ~M !TM denotes the di�erential of f . r naturally extends to a covariant derivative forvector �elds along f . For any vector �eld A on ~M and any vector �eld Y along f ,i.e. Y : ~M ! TM satis�es � �Y = f where � : TM !M denotes the projection, thecovariant derivative rAY is well de�ned. Due to the fact that (rAY )p depends onlyon Ap and the values of Y in a neighbourhood of the point p, this extension is uniquelydetermined by requiring the chain rule rv(Xf) = rf�vX for any tangentvector v � T ~Mand any vector �eld X on M .In a similar way the corresponding covariant derivative for tensor �elds carries overto a covariant derivative for tensor�elds along a map. As a consequence one obtainsfor example the Cartan structural equations for the Levi Civita connection:rAf�B � rBf�A � f�[A;B] = 0 (4)R(f�A; f�B)Y = rArBY � rBrAY � r[A;B]Y ; (5)where R is the curvature tensor of r, A, B are vector �elds on ~M and Y is a vector�eld along the map f .For a curve c : I !M the parameter vector �eld on I with respect to the parametert will be denoted by @@t or Dt , _c(t) = c� @@t jt is the the tangent vector of c at t. Thecovariant derivative rDtY for a vector �eld Y along c is abbreviated by Y 0 . A parallelvector �eld Y along c is characterized by the linear di�erential equation Y 0 = 0,a geodesic curve by the non-linear second order equation _c 0 = 0. For consistencyreasons we avoid the often found notation r_c _c resp. r_cY for the expressions rDt _cresp. rDtY when Y is a vector �eld along c. The inconsistency of such notationbecomes apparent when c is a singular curve for example a constant curve and Y anon-constant vector �eld along c. If X is a vector �eld on M, r_cX = rDtXc (chainrule) is well de�ned.The exponential map exp : TM !M is determined by the initial value problem forgeodesics. If v � TpM , then exp(v) = c(1) where c is the geodesic with initial conditionc(0) = p and _c = v . The restriction of exp to the tangent space TpM at p is denotedby expp . Notice that for complete manifolds the exponential map is de�ned on all ofTM by the Hopf-Rinow theorem.For a function f :M ! IR and a vector �eld X onM , Xf denotes the derivativeof f in direction X . The gradient of f is de�ned via the equationhgradf ;Xi = Xf (6)3

and the Hessian Hessf of f byHessf (X) = rXgradf : (7)Hessf is a selfadjoint endomorphism �eld, i.e.hrXgradf ; Y i = hrY gradf ;Xi.Important functions on a Riemannian manifold are distance functions or local dis-tance functions from some point in M or from a submanifold of M . A local distancefunction is a function in an open subset U of M considered as a Riemannian subman-ifold. If p 2 U � M and r(q) = distM(p; q), rU(q) = distU(q; p) then rU(q) � r(q).rU may be di�erentiable in points where r fails to be di�erentiable. A typical examplearises as follows: Let c : [�; �]!M be an injective geodesic segment with initial pointp = c(�) and without conjugate points. Then there is a neighborhood U of c(]�; �])where rU is di�erentiable. However r is not di�erentiable in any point of the cut locusof p. For explicit examples look at geodesics on a cylinder.On the set of points where a (local) distance function is di�erentiable it satis�esjjgradf jj = 1. The gradient lines of any function with this property are geodesicsparametrized by arc length, since Drgradf gradf ;XE = hHessf gradf ;Xi =hHessf X; gradf i = hrXgradf ; gradf i = 12 X hgradf ; gradf i = 0 for any vector �eldX on M and hence rgradf gradf = 0. Therefore the level surfaces of such a functionare equidistant. They are referred to as a family of parallel surfaces.1.2 Jacobi �eldsJacobi �elds J along a geodesic arise naturally as variational vector �elds in one pa-rameter families of geodesic lines and are characterized by the linear second orderdi�erential equation J 00 +R(J; _c) _c = 0: (8)If V is a geodesic variation of c, i.e. V : I � (�"; ") ! M is di�erentiable andV (t; 0) = c(t) and t 7! V (t; s) is a geodesic for all s � (�"; "), then J(t) = V� @@s jt;0 is aJacobi �eld along c:J 00(t) = rDtrDtV�Ds jt;0 = rDtrDsV�Dt jt;0 +rDtV� [Ds; Dt]| {z }=0 jt;0= rDtrDsV�Dt jt;0 �rDsrDtV�Dt| {z }=0 jt;0= �R(V�Ds; V�Dt)V�Dt jt;0 = �R(J; _c) _c t:4

Therefore the Jacobi equation is the linearization of the geodesic equation along c.Notice that V can be written in the following way: If p is the curve p(s) = V (0; s) andY the vector �eld along p given by Y (s) = V�Dt j0;s , then V (t; s) = exp tY (s). Theinitial conditions of the Jacobi �eld in terms of p and Y are J(0) = _p(0), J 0(0) = Y 0(0).Y (0) is the initial vector of the geodesic c. Any tangent vector u to TM can be writtenas the tangent vector u = _Y j0 of a curve s 7! Y js 2TM . Y is a vector �eld alongthe base curve p(s) = � � Y js . If Y and V are de�ned as above, we �nd exp� u =_dexp � Y j0 = V�Dsj1;0 = J(1). Therefore the di�erential of the exponential map iscompletely determined by Jacobi �elds.For example, the Jacobi �eld with initial conditions J(0) = 0, J 0(0) = w along thegeodesic exp tv is obtained from the variation V (t; s) = exp t(v + sw). Here p(s) isthe constant curve, Y (s) = v + sw , J(t) = exp� jtvtw , J(1) = expp �jvw . This showsthat the di�erential of the restriction expjTpM is determined by Jacobi �elds on Mwith these initial conditions.1.3 Interpretation of curvature in terms of the distance func-tionConsider two geodesics c0 , c1 emanating from a point p in M , c0(") = exp "v , c1(") =exp "w , v ,w 2 TpM and the distance L(") = dist(co("); c1(")) in a neighborhood ofzero. Then the fourth order Taylor formula for L2 is given byL2(") = "2jjv � wjj2 � 13 "4 hR(v; w)w; vi+O("5) : (9)When v 6= w this implies for " � 0 :L(") = "jjv � wjj � 16 hR(v; w)w; vijjv � wjj "3 +O("4) : (10)For linearly independent vectors v; w satisfying jjvjj = jjwjj = 1 this can be rewrittenas L(") = "jjv � wjj �1� 112K(v; w)(1 + hv; wi)"2�+O("4) ; (11)where K(v; w) ist the sectional curvature of the plane spanned by v and w . ThereforeL grows faster than linear if K < 0 and slower than linear if K > 0 in a neighborhoodof 0.To prove (9) we consider the variationV ("; t) = exp(t exp�1c0(") � c1("))5

p0v

w

K < 0

p0v

w

K ≡ 0

p0

v

w

K > 0Figure 1: interpretation of sectional curvaturep0

v

w

c1

c0

c1 (ε)

c0 (ε)

E(ε,t)

T(ε,t)

aε (t)

Figure 2: setup for the proof of (9)for small values of " and t2 [0; 1]. The parameter tangent �elds along V are E = V�D"and T = V�Dt . The parameter curves a" : t 7! V ("; t) are geodesics connecting thepoints c0(") and c1("). T is the tangent �eld of the geodesics and t 7! Ej";t is a Jacobi�eld along a" and Ej";0 = _c0("); Ej";1 = _c1(").Notice that jj _a"(t)jj is the length of a" so thatL(") = jj _a"(t)jj = jjT jj";t (12)which is constant in t for " �xed. The derivatives of H = L2 up to the fourth order6

are given byH 0(") = 2 hrD"T ; T i j";tH 00(") = 2 �Dr2D"T; TE+ hrD"T ;rD"T i� j";tH 000(") = 2 �Dr3D"T; TE+ 3 Dr2D"T;rD"TE� j";tHIV (") = 2 �Dr4D"T; TE+ 4 hr3D"T;rD"T i+ 3 Dr2D"T;r2D"TE� j";t :We will now evaluate these derivatives at (0; t) in order to �nd the coe�cients for theTaylor formula. The equation rD"T = rDtE and rDtT = 0 will be used frequentlyduring this calculation. Also notice that T j0;t = 0, since V (0; t) = p. We haverD"E j";0 = 0; rD"E j";1 = 0 (13)since Ej";0 = _c0(") and Ej";1 = _c1(") . From the Jacobi property of E we obtainrDtrDtE = �R(E; T )T ; (14)so that rDtrDtE j0;t = 0 : (15)Hence t 7! Ej0;t is a linear vector�eld along the constant curve a0 . Since Ej0;0 =_c0(0) = v , Ej0;1 = _c1(0) = w it followsEj(0;t) = v + t(w � v) : (16)With this information we can already evaluate H 0(0) and H 00(0):H 0(0) = 2 hrD"T ; T i j0;t = 0 (17)H 00(0) = 2 Dr2D"T; TE j0;t + 2 hrD"T ;rD"T i j0;t= 2 hrDtE;rDtEi j0;t= 2jjv � wjj2 (18)from (16). Next we show that rD"Ej0;t = 0 (19)rDtrD"Ej0;t = 0 (20)rD"rD"T j0;t = 0 : (21)(20) is a consequence of (19) and (21) follows from (20) sincerD"rD"T = rD"rDtE = R(E; T )E + rDtrD"E :7

In view of the equations (13) above it su�ces to show rDtrDtrD"E j0;t = 0 for theproof of (19). For this observerDtrDtrD"E = rDt(R(T;E)E) +R(T;E)rD"T + rD"(R(T;E)T ) :The right hand side vanishes at (0; t) since rDtT = 0 and T j0;t = 0. This su�ces to�nd H 000(0): H 000(0) = 6 hrD"rD"T ;rD"T i j0;t= 6 hrD"rDtE;rD"T i j0;t= 6 hR(E; T )E;rD"T i j0;t + 6 hrDtrD"E ;rD"T i j0;t= 0 (22)from (20). From (21) we getHIV (0) = 8 hrD"rD"rD"T ; rD"T i j0;t: (23)Furthermore rD"rD"rD"T j0;t = rD"rD"rDtEj0;t= (rD"R(E; T )E +rD"rDtrD"E)j0;t= R(E;rDtE)Ej0;t +rD"rDtrD"Ej0;t : (24)Using (16),(23),(24) and the symmetries of R we �ndHIV (0) = 8 hR(v; w)v; wi+ hrD"rDtrD"E ; rD"T i j0;t :Since this must also be constant in t, the second term on the right hand side is constantin t. Now hrD"rDtrD"E;rD"T i j0;t = D"Dt hrD"E;rD"T i j0;t= DtD" hrD"E;rD"T i j0;tby using (20) and (21). ThereforeD" hrD"E ; rD"T i j0;t = hrD"rD"E ; rD"T i j0;t + hrD"E ; rD"rD"T i j0;tmust be linear in t. But rD"rD"E j0;0 = rD"rD" _c0(0) = 0, rD"rD"E j0;1 =rD"rD" _c1(0) = 0 and rD"rD"T j0;t = 0, so that D" hrD"E ; rD"T i j0;t = 0 since itvanishes at t = 0 and at t = 1. This provesHIV (0) = 8 hR(v; w)v; wi : (25)8

Equation (9) now follows from (17), (18), (22) and (25). We leave it to the reader toverify HV (0) = 10 h(rv+wR)(v; w)v; wi : (26)If HV (0) = 0 for all choices of v and w , then M must be a locally symmetric space,since (26) can be used to show that the operator R _c = R(::; _c) _c is parallel for anygeodesic c.1.4 The levels of a distance functionIn this section we will see that Jacobi �elds determine the second fundamental tensorS of the level surfaces of a (local) distance function f . This will be used to establishthe Riccati equation for S and a Riccati inequality.We have a natural unit normal vector �eld N = gradf along the level surfaces off . The second fundamental tensor S of the levels with respect to N is the restrictionof the Hessian of f to the tangent spaces of the levels , Su = Hessf u = ruN fortangent vectors u to the levels. The derivative S 0 = rNS in the normal direction isde�ned by S 0Y = (rNS )Y = rN (SY ) � S(rNY ) for any vector �eld Y tangent tothe levels of f . Notice that S 0Y is again tangent to the levels.Let M0 be a �xed level, M0 = f�1f0g after changing f by a constant. The otherlevels are then given by Mt = f�1ftg. For small values of t the levels M0 and Mtare di�eomorphic via the di�eomorphism Et(p) = exp(t N(p)). The di�erential ofEtjMo can be desribed in terms of Jacobi �elds: Let s 7! p(s) be a curve in Mo withtangent vector v = _p(0). Then Et �v = J(t) where J is the Jacobi �eld along thegeodesic t 7! Et(p(0)) of the geodesic variation V (t; s) = Et � p(s), J(t) = V�Dsjt;0 . Itsinitial conditions are J(0) = _p(0) = v , J 0(0) = rDs(N � p)j0 = r_p(0)N = Sv , comparesection 1.2.The geodesic (t) = V (t; s) is an integral curve of N , so that V�Dtjt;s = _(t) =N � V (t; s). With this information we obtain J 0(t) = rDtV�Ds jt;0 = rDsV�Dt jt;0 =rDsN � V jt;0 = rV�DsN jt;0 = SJ(t). The second fundamental tensor of the levels nowis determined by SJ = J 0 : (27)Covariant di�erentiation of this equation leads to the important Riccati equation forS : Since (27) is an equation along the geodesic c(t)=V(t,0) it reads more preciselyScJ = J 0 . This is useful to remember for the chain rule in rDtSc = r_cS = rNcS =(S 0)c for the computation of J 00 = rDt(ScJ) = S 0J + SJ 0 = S 0J + S2J . Using the9

Jacobi equation J 00 +R(J;N)N = 0 we obtain the Riccati equationS 0 = �RN � S2 (28)where RN denotes the curvature operator RNX = R(X;N)N in direction N .If there is a lower bound � for the sectional curvature K of M , then the Riccatiequation leads to a Riccati inequality along the gradient lines c of f . Let Y be aparallel unit vector �eld along c tangent to the levels, i.e. hY; _ci = 0. Then by (28)hSY; Y i0 = �hR(Y;N)N; Y i � DS2Y; Y E= �K(Y;N)� jjSY jj2 :From the assumption � � K(Y;N) and the Schwarz inequality we obtain the Riccatiinequality hSY; Y i0 � ��� hSY; Y i2 (29)along c.1.5 Data in the constant curvature model spacesConstant curvature model spaces are important in comparison theory because thegeometric quantities in these spaces can be calculated explicitly.Mn� denotes the n-dimensional hyperbolic space IHn� of curvature � if � < 0, theeuclidian space IRn if � = 0 and the standard sphere Sn� of radius 1p� if � > 0.Since hR(v; u)u; vi = � for any pair of orthonormal vectors u; v2TpMn� , we have Ru :=R(:::; u)u = � � Idp on the orthogonal complement of u in TpMn� . Therefore the Jacobiequation and the Riccati equation are rather simple.Jacobi �elds along a geodesic c : IR!Mn� orthogonal to _c are given by f �Y , whereY is a parallel vector �eld along c and f : IR! IR is a solution of the 1-dimensionalJacobi equation f 00 + �f = 0 (30)Let sn� and cs� be the solutions of (30) with initial conditions sn� (0) = 0; sn� 0(0) = 1and cs� (0) = 1; cs� 0(0) = 0, i.e.sn�(t) = 1p� sinp�tcs�(t) = cosp� t 9=; for � > 0sn�(t) = tcs�(t) = 1 9=; for � = 0 (31)10

sn�(t) = 1pj�j sinhqj�j tcs�(t) = coshqj�j t 9>=>; for � < 0Furthermore let ct� (t) = cs� (t)=sn� (t) for sn� (t) 6= 0 (32)The derivatives of these functions are given bysn� 0 = cs� ; cs� 0 = �� sn� ; ct� 0 = ��� ct� 2 : (33)Furthermore the following elementary equations hold:1 = cs�2 + � sn�2 (34)sn�(a + b) = sn�(a)cs�(b) + cs�(a)sn�(b) (35)cs�(a + b) = cs�(a)cs�(b)� � sn�(a)sn�(b) : (36)A basis for the Jacobi �elds orthogonal to _c is given by fsn� � Y; cs� � Y g where Yvaries over a basis of parallel vector �elds orthogonal to _c .Notice that the second fundamental tensor of the (local) distance spheres at distancer from a �xed point p in any manifold is determined by equation (27), where J is aJacobi �eld with initial value J(0) = 0 along a normal geodesic emanating from p, i.e.in Mn� by J(r) = sn� (r)Y (r)with Y parallel along c and hY; _ci = 0. HenceSc(r)Y (r) = sn� 0(r)sn� (r) Y (r) = ct� (r)Y (r) : (37)Therefore the principal curvatures of distance spheres in Mn� are equal to ct� (r).The length of the great circles in the distance spheres is 2� sn� (r). In any manifoldthe Hessian of the distance function from a point has a zero eigenvalue in the radialdirection. For Karcher's new proof of Toponogov's theorem it is convenient to rescalethe distance function f from the point p in Mn� so that all the eigenvalues are equal.This is achieved by taking md� � f , wheremd�(r) = Z r0 sn� (t)dt = 8

From the formulaHess(md��f)v = (md�0�f)Hessf(v) + (md�00�f) hgradf; vi gradf= (sn��f)Hessf(v) + (cs��f) hgradf; vi gradf (40)it follows that the eigenvalues of Hess(md� � f) at a point q with f(q) = r are equalto cs� � f(q) = cs� (r). Using md� , the law of cosines in Mn� becomesmd� (c) = md� (a� b) + sn� (a)sn� (b)(1� cos ) (41)where a; b; c are the lengths of the edges of a geodesic triangle in M� and is theangle opposite to the edge corresponding to c. Notice that this is a uni�ed formula forthe three classical cases � = 0,� > 0,� < 0:c2 = a2 + b2 � 2ab cos (42)cos(p� c) = cos(p� a) cos(p� b) + sin(p� a) sin(p� b) cos (43)cosh(qj�j c) = cosh(qj�j a) cosh(qj�j b)� sinh(qj�j a) sinh(qj�j b) cos (44)1.6 The Riccati comparison argumentA lower curvature bound � in M leads to an important estimate for the principalcurvatures in distance spheres and hence for the tangential eigenvalues of the Hessianof the distance function f from a point. For the modi�ed distance function md� � fthis yields an estimate for all the eigenvalues. This estimate is the key for Karchersproof of Toponogov's theorem and the main reason for introducing md� . The basiccomparison argument is contained in (i) of the following elementary Lemma and itsCorollary, cf. [K].Lemma 1.1 Suppose g , G are di�erentiable functions on some interval satisfying theRiccati inequalities g0 � ��� g2 (45)G0 � ���G2 : (46)i) If g(r0) � G(r0), then g(r) � G(r) for r � r0 .ii) If g(r0) � G(r0), then g(r) � G(r) for r � r0 .12

Proof. From the two Riccati inequalities (45) and (46) we get[(g �G) � eR (g+G)]0 � 0from which i) and ii) follow immediately. 2The statement ii) is useful for estimates involving upper curvature bounds [K]. Weare interested mainly in i).Corollary 1.2 If g : (0; a) ! IR (suppose a � �p� if � > 0) satis�es g0 � �� � g2and limr!0 g(r) =1, then g(r) � ct� (r) :Proof. If there is a point r02(0; a) for which g(r0) > ct� (r0), we can choose " > 0 sothat g(r0) � ct� (r0�"). G(r) = ct� (r�") satis�es the Riccati equation G0 = ���G2on ("; r0), so that g(r) � G(r) on ("; r0). Then g(") = limr&" g(r) � limr&"G(r) =+1, contradicting g(") 0 appears at a distance not farther away than �p� . For M = M2� equalityholds in (47).If f is replaced by g = f + � where � is a constant, we have Hess g = Hessf sothat the tangential eigenvalues of Hess(md� � g)jq according to formula (40) are givenby sn� (g(q))�i(q) and the radial eigenvalue is cs� (g(q)). The estimate for �i(q) above13

leads to (sn� � g)�i � (sn� � g)ct� (g � �) = cs� � g + sn� (�)sn� (g��) . For small values of �and 0 < g � � < �p� in the case � > 0 the Hessian of md� � g satis�es consequentlyHess(md� � g) � (cs� � g + sn� (�)sn� (g � �)) � Id : (48)In the case � > 0 this estimate along c holds up to the �rst conjugate point.2 The Toponogov TheoremThe Toponogov comparison theorem appears to be one of the most powerful toolsin Riemannian geometry. It is a global generalization of the �rst Rauch comparisontheorem. The ideas trace back to A.D. Alexandrow who �rst proved the theorem forconvex surfaces. Toponogov's proof of the theorem was technical and contained somedi�culties which were resolved in [GKM]. Since then the proof had been simpli�edconsiderably by various geometers, compare also [CE]. In this lecture series we shalluse an interesting new proof given by Karcher [K]. In contrast to the previous techniquethe Rauch comparison theorem is not used at all. It uses the estimate for the Hessiangiven in (47) resp.(48) and �ts nicely into our discussion of distance functions. Thisdoes not mean, that our approach is necessarily shorter or more geometric than theother viable arguments given before. We certainly encourage the student also to gothrough some alternate proof of Toponogov's basic result in the literature mentionedabove.De�nition 2.1 A geodesic hinge c; c0; � in M consists of two non constant geodesicsegments c,co with the same initial point making the angle �. A minimal connectionc1 between the endpoints of c and co is called a closing edge of the hinge.The length of a geodesic segment c will be denoted by jcj.Theorem 2.2 (Toponogov) Let M be a complete Riemannian manifold with sec-tional curvature K � �.A) Given points p0; p1; q in M satisfying p0 6= q; p1 6= q , a non constant geodesic cfrom p0 to p1 and minimal geodesics ci , from pi to q ,i = 0; 1, all parametrized byarc length. Suppose the triangle inequality jcj � jc1j+jc2j is satis�ed and jcj � �p�in the case � > 0. �i2[0; �] denote the angles at pi , �0 =

i) the corresponding angles ~�i satisfy ~�i � �iii) dist(~q; ~c(t)) � dist(q; c(t)) for any t2 [0; jcj] .Except for the case when � > 0 and one of the geodesics has length equal to �p�the triangle in M2� is uniquely determined.B) Let c; co; �o be a hinge in M with co minimal and jcj � �p� in case � > 0 andc1 a closing edge . Then the closing edge ~c1 of any hinge ~c; ~co; �o in M2� withj~cj = jcj , j~coj = jcoj satis�es j~c1j � jc1j :Remarks1. Notice that c need not to be minimal and the case p0 = p1 is not excluded. c1and c0 have to be minimal, otherwise there are counterexamples.2. With a little e�ort statement (ii) can be used to show that the length of secantsbetween c and ci are not shorter than the corresponding secants between ~c and~ci , provided the segment of c in the cut o� triangle is minimal:iii) dist(~c0(t); ~c(s)) � dist(c0(t); c(s)) holds as long as cj[0;s] is minimal,iv) dist(~c1(t); ~c(s)) � dist(c1(t); c(s)) holds as long as cj[s;jcj] is minimal.In the case when c is minimal now any corresponding secants �; ~� satisfy j~�j �j�j.For symmetry reasons only iii) needs to be proved:By ii) dist(~q; ~c(s)) � dist(q; c(s)) : (49)Connect ps = c(s) and q by a minimal geodesic s and consider the trianglep0 ,ps ,q with geodesic edges c0 , cj[o;s] ,s and the corresponding comparison tri-angle ~p0; ~ps; ~q in M2� . Using ii) for this triangle we obtaindist(~ps; ~c0(t)) � dist(c(s); c0(t)) : (50)The monotonicity relation between angle and length of the closing edge of a hingein M2� and (49) imply

p̃0

c̃0 (t)

p̃s

c̃(s)

q̃

p̃1

c̃1

Mκ2

p0

c0 (t)

c(s) = ps

q

p1

c1

M

Figure 3: sketch for the proof iii)and then dist(~c(s); ~c0(t)) � dist(~ps; ~c0(t)) : (51)Inequality iii) now follows from (50) and (51).3. Statement i) is a consequence of ii). To prove for example �0 � � , consider thefunctions h0(t) = dist(c0(t); c(t))2 and ~h0(t) = dist(~c0(t); ~c(t))2 for small valuesof t. By iii) we have ~h0 � h0 . According to (9) of section 1 we have the Taylorformulas h0(t) = t2k _c0(0)� _c(0)k2 +O(t4)~h0(t) = t2k_~c0(0)� _~c(0)k2 +O(t4)so that k_~c0(0)� _~c(0)k � k _c0(0)� _c(0)k and hence ~�0 � �0 .The converse implication i) ) ii) is also true but more technical to prove.4. Statement ii) carries over to limits in the sense of Gromov for Riemannian spaceswith curvature K � �, where angles cannot be de�ned anymore.5. Part B is equivalent to A)i). This follows immediately from the fact that in M2�the length of a closing edge in a hinge with minimal geodesics and the hinge angleare in a monotone relation. Note that B is trivial in the case when the tiangleinequality is not satis�ed in M . For this observe that the triangle inequality inM2� is satis�ed since all the corresponding geodesics in M2� are minimal.6. If c0 is not minimal in B), the statement is false. For consider in S21+" a hingewith two geodesics of length � making a positive angle. The end points have a16

positive distance for " small. However, in the corresponding hinge in S21 the endpoints coincide.7. An anologue of Toponogov's theorem where the lower curvature bound is replacedby an upper curvature bound is false. For example on the 3-sphere S3 there arehomogeneous metrics (Berger metrics) with positive curvature, upper curvaturebound 1 and closed geodesics of length < 2� . However, if the sectional curvatureK of M satis�es K � � and c0 ,c1 ,c is a triangle with minimal geodesics andjc0j+ jc1j+ jcj < 2�p� which is contained in a ball around p0 of radius not greaterthan the injectivity radius at p0 , then there is a triangle ~c0 , ~c1 , ~c in M2� withjcij = j~cij; jcj = j~cj and �0 � ~�0: This is an immediate consequence of Rauch's�rst comparison theorem.8. There are generalisations of Toponogov's theorem to a version where the modelspaces M2� are replaced by surfaces of revolution or surfaces with an S1 - action,c.f. [E], [A]. U. Abresch pointed out to me that these generalisations can behandled with the same technique as used in the proof below.Proof of Theorem 2.2. By remark 2 above we only have to prove A)ii). Note thatin the case � > 0 we have diam(M) � �p� by Myers' theorem. For the case � > 0the proof is organized in three steps. In step 1 we consider the general case for � � 0,but we assume diam(M) < �p� and jcj + jc0j + jc1j < 2�p� for the case � > 0. In step2 the case � > 0, diam(M) � �p� and jcj + jc0j + jc1j � 2�p� is reduced to step 1 by asimple limit argument. Finally, in step 3 we show that in the case � > 0 there are notriangles with circumferece jcj+ jc0j+ jc1j > 2�p� .Step 1. For the case � > 0 we assume diam(M) < �p� and also that the circumferenceof the triangle satis�es jcj + jc0j + jc1j < 2�p� , so that the comparison triangles in M2�exists. From the triangle inequality jcj � jc0j + jc1j we get jcj < �p� for � > 0.Therefore we can choose " > 0 such that diamM < �p� � 2" and jcj < �p� � 2". We�rst look at a simple case: Suppose q2c(]0; jcj[). Then jcj � jc0j + jc1j since c1 andc0 are minimal. By the triangle inequality we must have jcj = jc0j+ jc1j. Therefore qdivides c into two minimal pieces of length jc0j and jc1j. Consequently equality holdsin ii) since the geodesics from q to c(t) are parts of c. If q 62c(]0; jcj[) we proceed asfollows:We consider the distance functions r from q in M , ~r from ~q in M2� and de�neh(t) = md��r�c(t)~h(t) = md�� ~r�~c(t)17

�(t) = h(t)� ~h(t) :The idea is, to show that � cannot have a negative minimum by the use of the Hessianestimate (48) in section 1.6. Unfortunately h is not di�erentiable in general since ris not di�erentiable beyond the cutlocus of q . This problem is resolved by a localapproximation with a \superdistance function". The argument is slightly di�erent inthe cases � < 0; � = 0 and � > 0.In the case � = 0, if � has a negative minimum �2� in ]0; jcj[ also the function ��de�ned by ��(t) = �(t) + �t(jcj � t)jcj2has a negative minimum < �� in ]0; jcj[ .In the case � > 0 we have jcj � �p� � 2" and de�ne �"(t) = sn� (t+ ")� sn� ( "2) on[0; jcj]. If � has a negative minimum then�̂ = ��"has a negative minimum.For the point t02]0; jcj[ where � or �̂ or �� has a negative minimum, we approximater by local di�erentiable functions in a neighborhood of c(t0). Let be a normalgeodesic from q to c(t0). For small values � > 0 we de�ne in some neighborhood Uof (]�; jj[) the local superdistance functionsr�(x) = � + distU((�); x) � r(x) = dist(q; x) :r� is di�erentiable if U is su�ciently small. Therefore the functionh� = md� � r� � c (52)is di�erentiable in some interval around t0 andh�(t0) = h(t0); h� � h : (53)Using the estimate (48) for the Hessian we haveh00� = hHess(md��r�)jc _c; _ci� cs��r��c+ sn�(�)sn�(r��c� �)18

for � small. The quantity r� � c(t) � � is bounded away from zero independent of �and r� � c(t)�� = dist((�); c(t)) � �p� �2" from the diameter assumption. Observing(39) we get h00� + �h� � 1 + const � sn� (�)with a constant independent of � . Since ~h00 + �~h = 1 the di�erence �� = h� � ~hsatis�es �00� + ��� � const � sn� (�) : (54)Furthermore �� � �; ��(t0) = �(t0) (55)by (53).Case 1. � < 0If � has a negative minimum �� at t0 , then �� also has a negative minimum �� att0 , but �00�(t0) � ���(t0) + const � sn� (�) = ��|{z} 0At the point t0 where �̂ = ��" has a negative minimum ��o we also look at �̂� = ���" .Again �̂� � �̂ and �̂(t0) = �̂�(t0) so that �̂� has a negative minimum ��0 at t0 .Di�erentiate at t0 to obtain 0 = �̂0�(t0) = �0��" � ���0"�2" jt019

and �̂00�(t0) = 1�2" (�"�00� � �00"��)jt0= 1�2" ((�00� + ���)�" + ���sn�("2))jt0� 1�"(t0)const � sn�(�)� ��0�"(t0)sn�("2) < 0for � su�ciently small, a contradiction.Step 2. Assume now � > 0, diam(M) � �p� and jcj + jc0j + jc1j � 2�p� . Wechoose a sequence �i , 0 < �i < � and limi!1 �i = �. Then diam(M) < �p�i andjcj+ jc0j+ jc1j < 2�p�i . By step 1 the theorem holds for the sphere S2�i � IR3 as the com-parison space. By compactness, the sequence of comparison triangles ~�i = (~ci; ~ci0; ~ci1)has a subsequence converging to a comparison triangle ~� in S2� . By continuity of thefamily of distance functions on the family of spheres S2~� � IR3 , ~� > 0, statement A)ii)now follows for the limit triangle ~�.Step 3. Suppose � > 0 and jcj + jc0j + jc1j > 2�p� . We can choose � > 0 such thatjcj + jc0j + jc1j = 2�p� Then for the comparison triangle in M2� the geodesics ~c0; ~c1; ~chave length < �p� and therefore form a great circle. The antipodal point �q of ~q hasto be a point of ~c, say �q = ~c(t0). By step 1 we have �p� = dist(~q; ~c(t0)) � dist(q; c(t0))contradicting dist(q; c(t0)) � �p� < �p� . This completes the proof. 2

20

3 Applications of Toponogov's Theorem3.1 An estimate for the number of generators for the fun-dametal groupAs a �rst application of Toponogov's theorem we present Gromov's theorem concerningthe number of generators for the fundamental group �1 (M). Since any element of thefundamental group �1 (M) with base point p of a Riemannian manifold M can berepresented by a geodesic loop of minimal length at the point p, it is clear that thegeometry of M should have strong inuence on the structure of �1 (M). The earliestresult in this direction is Myers' theorem, cf.[CE], [GKM]: the universal cover of acompact Riemannian manifold with strictly positive Ricci curvature is compact and thefundamental group �nite. If the sectional curvature K of a compact even dimensionalmanifold is strictly positive, then by the Synge Lemma, cf. [CE], [GKM], �1 (M) = 1or Z2 depending on the orientability of M . If M is complete non-compact and K > 0,then �1 (M) = 1 since M is di�eomorphic to IRn , cf. [GM]. Finally if M is completenon-compact and K � 0, then by the soul theorem of Cheeger and Gromoll [CG1],�1 (M) contains a lattice group of �nite index.Theorem 3.1 (Gromov)(i) Suppose the sectional curvature of Mn is nonnegative. Then �1 (Mn) can begenerated by N � p2n� 2n�2 elements.(ii) If the sectional curvature K of Mn is bounded from below, K � ��2 and thediameter of Mn is bounded, diamMn � D , then �1 (Mn) can be generated byN � 12p2n�(2 + 2 cosh(2�D))n�12 elements.Proof. Let G = �1 (M; p0) be the fundamental group with base point p02M . ~Mdenotes the Riemannian universal cover of M . The group of covering transformationsG acts on ~M by isometries. We choose a point x02 ~M which covers p0 and de�ne for2G the displacement jj := dist(x0; x0)A minimal geodesic c from x0 to x0 projects in M to a loop of minimal length jjin the homotopy class representing . There are only �nitely many elements of Gsatisfying jj � r . (An in�nite sequence ix0 of points would have a limit point in thecompact ball of radius r around zero contradicting the covering property.) Thereforewe can choose an element 12G with the property j1j = minfjj j 2Gg. Inductively21

we can construct generators 1 , 2 , ... of G satisfying j1j � j2j � ::: as follows:Suppose 1; :::; k are constructed already and the subgroup < 1; :::; k > generatedby 1; :::; k is not equal to G. Then we can choose k+12G so that jk+1j = minfjj j2Gn < 1; :::; k >g . For i < j we have jij � jjj and`ij := dist(ix0; jx0) � jjj :To prove the last inequality, suppose `ij < jjj. Then 0j := �1i j has displacementj0jj = `ij < jjj and < 1; :::; j >= < 1; :::; j�1; 0j >, contradicting the choice ofj .For each i we choose a minimal geodesic ci from x0 to ix0 of length `i = jij. Fori < j we choose a minimal geodesic from ix0 to jx0 of length `ij . By Toponogov'stheorem the angle �ij =

then �ij � ~� � �� . To complete the argument consider the initial vectors vi =_ci(0)2Tx0 ~M . We have 0. In Tx0 ~M there can be only a �nite numberof distinct unit vectors with this property. A rough explicit estimate for the maximalnumber is obtained as follows: The intrinsic balls of radius ��=2 around the pointsvi in the unit sphere Sn�1 in Tx0 ~M are disjoint. Therefore the maximal number N�of points vi is estimated by the volume of Sn�1 divided by the volume of a ball ofradius ��=2 in Sn�1 . The volume of this spherical ball is estimated below by thevolume of a euclidian (n-1) ball of radius sin(��=2). This estimate, however, can beimproved by a factor 12 by the following simple observation: The generators satisfyjij = j�1i j. Therefore we also have

distA , if for any vector v2TqM there is a distance minimizing geodesic c from q to Asatisfying hv; _c(0)i � 0 : (61)A non-critical point is called a regular point.For points q 62A this condition is equivalent to 0for any minimal geodesic c from q to A. 24

q2 q2q3

q1p

q1

q2 q3 q2

cut locus

flat torus

q1

p

q3

q2cut locus

torus of revolutionFigure 5: critical points and cut locus for a point p in toriLemma 3.3 (local existence of gradient-like vector �elds) Let M be completeand A a closed subset of M . Then for any regular point q of distA there is a unitvector�eld X on some open neighborhood U of q such thathX~q; _c(0)i < 0 (62)for any ~q2U and any minimal geodesic c from ~q to A.De�nition 3.4 A unit vector �eld X on U satisfying (62) is called a gradient-likevector �eld for distA .Proof. Since q is a regular point we can choose a unit vector Xq2TqM with hXq; _c(0)i

Proof. a) is obvious from (62). For the proof of b) we point out that local vector �eldsof the lemma can be glued together by means of a partition of unity to obtain a vector�eld ~X on U satisfying (62). This is a consequence of the following observation: Ifv1; : : : ; vm are unit vectors in a euclidian vector space satisfying hvi; wi < 0, then anyconvex linear combination v = Pmi=1 �ivi , �i � 0, Pmi=1 �i = 1 satis�es hv; wi < 0.Now we can take X = ~X=k ~Xk. 2The following lemma contains an important monotonicity property for gradient-likevector �elds.Lemma 3.6 Let M be complete, A a closed subset, U an open subset of M and Xa gradient-like vector �eld for distA on U , � the ow of �X and the ow of X .Thena) distA is strictly decreasing along any integral curve of �X .b) On any compact subset C of U the decreasing rate is controlled by a Lipschitzconstant: There is a constant � > 0 such thatdistA�(q; t0 + �) � distA�(q; t0)� �� (63)as long as �(q; t0 + �)2C for 0 � � � � . Equivalently we havedistA(q; t00 + �) � distA(q; t00) + �� (64)as long as (q; t00 + �)2C for 0 � � � � .Proof. It su�ces to prove b). First notice that X satis�es the inequality h�Xq; _c(0)i �� for some � > 0, any q2C and any minimal geodesic c from q to A. Other-wise there would be sequences qi2C and minimal geodesics ci from qi to A withlimi!1 hXqi; _ci(0)i � 0. By compactness there would be a limit point q2C and aminimal limiting geodesic from q to A with hXq; _c(0)i � 0, contradicting (62). Con-sider now the function h(t) = distA�(q; t). We construct an upper support function~h for h as follows: Let p2A with dist(p;�(q; t0)) = distA�(q; t0) and choose a min-imal geodesic c : [0; 1] ! M from �(q; t0) to p . For a �xed � , 0 < � < 1 let~h(t) = � + dist(c(jcj � �);�(q; t)). ~h is di�erentiable in a neighborhood of t0 and sat-is�es ~h(t) � dist(p;�(q; t)) � distA�(q; t) = h(t) and ~h(t0) = h(t0). The derivativeat t0 is given by ~h0(t0) = Dgrad distc(jcj��)j�(q;t0);�� @@t jq;t0E = h� _c(0);�X ��(q; t0)i,hence ~h0(t0) � ��. Such a support function exists at any t0 , therefore condition (63)26

follows easily. 2As an immediate consequence we have:Corollary 3.7 Any local maximum point q of distA is a critical point for A.Corollary 3.8 Let Mn be complete and B = B(p; r) a ball of radius r around thepoint p2M . Suppose there are no critical points of distp in @B .Then @B is a topological (n-1)-submanifold of M .Proof. We only have to show that @B is locally euclidian. Consider a vector �eldX on the set of regular points with property (62) and the ow � of �X . For agiven point q2@B let Q be a local (n-1)-dim submanifold through q which is transver-sal to X , for example the image under the exponential map of a neighborhood Vof the origin in the (n-1)-plane orthogonal to Xq in TqM . By the inverse map-ping theorem we can assume that V and " > 0 are chosen such that �jQ�[�";"]is a local di�eomorphism. Since t 7! distp�(q; t) is strictly decreasing, we havedistp�(q; ") < distp�(q; 0) < distp�(q;�"). Therefore by continuity we can assumethat distp�(~q; ") < distp�(q; 0) < distp�(~q;�") for ~q2Q, after shrinking Q if neces-sary. Now any integral curve of X through a point ~q of Q meets exactly one point of@B = dist�1p (r) by the monotonicity property. The map from Q to @B de�ned by theprojection along the integral curves is a homeomorphism onto its image. 2Corollary 3.9 Let M be a complete non-compact manifold and suppose that for somepoint p2M all the critical points of distp are contained in a ball B = B(p; r)Then M is homeomorhpic to the interior of a compact manifold with boundary.Proof. Let X be a gradient-like vector �eld on the set of regular points with ow .Then F : @B � [0;1[! M de�ned by F (q; t) := (q; t) maps @B � [0;1[ homeo-morphic onto M n B . For this the properties (64) and kXk = 1 < 1 are important.Hence M is homeomorphic to B[F (@B� [0;1[) � B[F (@B� [0; 1[) with boundaryF (@B � f1g) � @B . 2Corollary 3.10 Suppose that there is no critical point of distp in B(p; r) n fpg.Then B(p; r) is contractible. 27

Proof. An easy exercise. 2De�nition 3.11 An isotopy of M (in the topological category) is a homotopy G :M � [0; 1]!M such that p 7! G(p; �) is a homeomorphism from M onto a subset ofM for any � 2[0; 1] and p 7! G(p; 0) is the identity map of M .If B1 and B2 are subsets of M , we say that the isotopy G moves B2 into B1 , providedG(B2 � f1g) � B1 .Corollary 3.12 (Isotopy Lemma) Given a complete manifold M , a point p2M ,0 < r1 < r2 � 1 and an open neighborhood U of the annulus A = B(p; r2) nB(p; r1).Assume that there are no critical points of distp in A.Then there is an isotopy of M which is the identity on M nU and which moves B(p; r2)into B(p; r1).Proof. Using a partition of unity one can construct a vector �eld X on M which isgradient-like on some neighborhood W of A with W � U and XjMnU = 0.If r2 < 1 we can choose � > 0 such that (62) holds on the compact set A. Thenfor t0 > 1�(r2� r1) the isotopy G de�ned by G(q; �) = �(q; � � t0) moves B(p; r2) intoB(p; r1), where again � is the ow of �X .If r2 = 1, we consider F : @B�] � 1;1[! M , F (q; t) = �(q; t) and use on thedomain of this homeomorphism onto a subset of M an isotopy induced from a defor-mation of ]�1;1[ into ]0;1[, for instance G(t; �) = ln(� + et). 2The elementary corollaries above demonstrate that gradient-like vector �elds canbe used for deformations in the same way as gradient vector �elds in standard Morsetheory. However all these deformation arguments are useless unless one can get addi-tional information on the set of critical points. In standard Morse theory the MorseLemma is an important tool for this purpose. Unfortunately, there is no analogue ofthe Morse Lemma available. In fact one cannot say much about the change of topologyof B(p; r) = dist�1p (r) when r passes a critical level.In the presence of a lower curvature bound, however, Toponogov's comparison theo-rem can be used to obtain additional information about critical points leading to ratherstrong conclusions. In contrast to standard Morse theory the information obtained onthe set of critical points is more of a global nature. For the proof of 3.19 one hasto consider not just a single distance function but all the distance functions from thevarious points of M . 28

3.3 The diameter sphere theoremOne of the famous results in Riemannian geometry is the 14 - pinching sphere theoremcf. [GKM], [CE], which can be stated as follows:Theorem 3.13 (Rauch, Berger, Klingenberg) Suppose Mn is complete, simplyconnected and the sectional curvature K satis�es14 < K � 1 :Then M is homeomorphic to the standard sphere.One of the essential steps in the proof of this theorem is to show that the injectivityradius of the exponential map and hence the diameter of M is � � > �2p� , where� > 14 is the minimum of the sectional curvatures on M . Grove and Shiohama havegeneralized the 14 -pinching sphere theorem to the diameter sphere theorem below byreplacing the upper curvature bound by this lower bound for the diameter. The proofis a nice application of critical point theory and of Toponogov's theorem.Theorem 3.14 (Grove-Shiohama) Let Mn be a complete manifold with K � � > 0and diamM > �2p� . Then M is homeomorphic to Sn .Proof. After rescaling the metric we can assume K � 1 and diamM > �2 . Let p, q betwo points of maximal distance in M , dist(p; q) = diamM . By corollary 3.7 q is criticalfor p. We show that q is uniquely determined by p. Suppose q1 , q2 are two pointssatisfying dist(p; qi) = diamM . Choose minimal geodesics c from q1 to q2 and ci fromqi to p. Since q1 is critical for p, c1 can be chosen such that �1 = �2 and ` := jcj � `1 . Consider the correspondingcomparison triangle ~c, ~c1 ~c2 in the standard sphere with corresponding angle ~�1 andedge lengths j~cj = `, j~c1j = j~c2j = `1 . Then by 2.2 ~�1 � �1 � �2 . By the law of cosinesin S21 we have0 � sin `1 sin ` cos ~�1 = cos `1 � cos `1 cos ` = (1� cos `) cos `1 � 0and hence ` = 0, i.e. q1 = q2 .Next we show that p and q are the only critical points for p, more precisely: Letq1 := q , q22M , q1 6= q2 6= p and c : [0; 1]!M be a minimal geodesic of length ` fromq1 to q2 . Then for any minimal geodesic c2 from q2 to p we have h _c(1); _c2(0)i > 0,i.e. the vector v := � _c(1)2Tq2M can be used to de�ne the open half space for the29

regularity of q2 . To show this, choose a minimal geodesic c1 from q1 to p and let`1 = jc1j, `2 = jc2j. By the uniqueness of q = q1 we have 0 < ` < `1 and 0

If one relaxes the assumption in the 14 -pinching theorem to 14 � K � 1, then thereis the rigidity theorem of Berger, cf. [CE]. In view of this result one also should expecta rigidity theorem if one assumes K � 1 and diamM = �2 . In fact Gromoll and Grovecf. [GG] have obtained a corresponding result:Under the given hypothesis eithera) M is homeomorphic to a sphere, orb) M has the cohomology ring of the Cayley plane, orc) M is isometric to one of the following spaces with their standard metrics: lCPm ,IHP` , lCP2d�1=f[z1; : : : :z2d] � [zd+1; : : : ; z2d;�z1; : : : ;�zd]g, Sn1 =�, where the or-thogonal representation of � = �1 (M) on IRn+1 is reducible.The proof is somewhat technical for our exposition.3.4 A critical point lemma and a �niteness resultThe critical point lemma below was one of the basic observations which lead Gromovto the �niteness theorem in the next section. Its proof is a simple application ofToponogov's theorem (used twice). The given estimate is somewhat stronger than inGromov's original lemma. It was also used by Abresch [A].Lemma 3.15 (critical point lemma) Let M be complete and p, q1 , q2 2M , qi 6= pand assume q1 is critical for p. Furthermore let ci be minimal geodesics from p to qiof length `i ,`1 � `2 and � = 0) and diamM < D , thencos� � `1`2 �D coth(�D) :Proof. Let c be a minimal geodesic from q1 to q2 of length `. Since q1 is critical for pthere is a minimal geodesic c1 from q1 to p of length `1 such that �1 =

Consider now the geodesic triangle c1 , c2 , c and the corresponding triangle ~c1 , ~c2 ,~c with the same edge length in the comparison space IR2 respectively M2��2 . Thenthe angle comparison theorem 2.2 A (i) leads to ~� = 1. Then thereare only �nitely many critical points q1; : : : ; qk for the distance function distpsatisfying distp(qi+1) � L � distp(qi) :If L � 3(1 +p2)n�1 , then k � 2n.b) For manifolds with K � ��2 and diamM < D the same statement holds forL � 3(1 +p2)n�1�D coth�D .RemarkBy reversing the indexing of the points qi we also have at most 2n critical pointssatisfying distp(qi+1) � 1Ldistp(qi)if L is chosen as speci�ed in the corollary.Proof of corollary 3.16. We consider the case K � 0 and leave the simple modi�-cation for b) to the reader. Connect p and qi by minimal geodesics ci of lengths `i .Then `i � L`j for i > j . By the critical point lemma the angles �ij = 0. There are only �nitelymany vectors in TpM with this condition, compare also the proof of theorem 3.1. IfL = 2 we have �ij � �3 and k � p2�n 2n�2 . If L � 3(1+p2)n�1 , then �ij � �2 � �n ,32

where �n = arcsin 13 � 11+p2�n�1 . By the ball packing argument due to Abresch, cf. [A]part II, there are at most 2n vectors in IRn making a pairwise angle � �2 � �n . 2Corollary 3.17 Let Mn be a complete non-compact manifold with K � 0, p2M .Then all critical points of distp are contained in some ball of �nite radius around p.As a consequence we obtain the followingTheorem 3.18 (Gromov) Let Mn be a complete non-compact manifold with K � 0.Then M is homeomorphic to the interior of a compact manifold with boundary, henceM is of "�nite" topological type.Proof. Since the critical points of distp are contained in some ball of �nite radius,corollary 3.9 applies. 2Recent examples of Sha and Yang show that a similar result does not hold formanifolds with positive Ricci curvature, cf. [SY1]. However if Ric > 0 and in additionK > �1 and the "diameter growth" of @B(p; r) is of the order o(r 1n ), then the sameconclusion as in the theorem holds, cf. [AG].The above theorem 3.18 may be viewed as a weak version of the much more subtlesoul theorem of Cheeger and Gromoll, by which M contains a compact totally geodesicsubmanifold S such that M is di�eomorphic to the normal bundle of S in M , cf.[CG1], [CE] and section 3.6.3.5 An estimate for the sum of Betti numbersIn this section H�(M) denotes the singular homology of M with coe�cients in somearbitrary �eld F . The kth Betti number of M with respect to F is given by bk(M) =dimF Hk(M). For a compact n-manifold and by theorem 3.18 also for complete n-manifolds M of nonnegative curvature we have Pnk=0 bk(M) = dimF H�(M)

b) Given D > 0 and � < 0 and n, then there is a constant C�(D; �; n) such that anycomplete n-manifold with sectional curvature K � � and diamM � D satis�esdimF H�(M) � C�(D; �; n) :In his paper [G2] Gromov indicated that a similar theorem as (a) holds for manifoldswith assymptotically nonnegative curvature. U. Abresch [A] gave the precise de�nitionof "assymptotically nonnegative curvature" for which such a theorem can be proved.He also re�ned Gromov's method and developed the necessary tools to obtain thefollowing result:c) Let � : IR+ ! IR+ be a decreasing function satisfying R10 r�(r) dr � 1. thenthere is a constant C#(n; �) such thatdimF H�(M) � C#(n; �)for any complete Riemannian manifoldMn with sectional curvatures Kr � ��(r)at distance r from a given point p2M .Remarks1. The lower bound for the sectional curvature cannot be replaced by a lower boundfor the Ricci curvature: Sha and Yang recently have constructed metrics of pos-itive Ricci curvature on the connected sum of an arbitrary number of copies ofSn�Sm , cf. [SY2]. In [SY1] they also gave complete noncompact examples withpositive Ricci curvature of in�nite homology type.2. Under the hypothesis in the theorem one cannot expect �niteness for the numberof homotopy types. Here the lens spaces and also the simply connected Wallachexamples [AW] should be observed.However Grove and Petersen have shown that there are only �nitely many homo-topy types of compact manifolds if in addition to the lower curvature bound andthe upper diameter bound one assumes a lower bound for the volume, cf. [GP].3. The methods for the proof of a) and b) are essentially the same. For the proofof c) Abresch had to develop a more general version of Toponogov's trianglecomparison theorem, compare remark 8 in section 2. Though the proof of c) issomewhat more technical, the re�ned method of Abresch leads to a simpli�edproof of a) and b). It also gives a better estimate for the constants C(n) andC�(n; �;D) than in [G2]. 34

For reasons of exposition we concentrate on the proof of a) using the re�ned versiondue to Abresch. So we assume K � 0 for the remainder of this section unless statedotherwise. We also will �x the constantL = 3(1 +p2)n�1as determined in corollary 3.16.It is convenient to use the the following notation in connection with metric balls:If B is a ball of radius r around p then �B denotes the concentric ball of radius �raround p.In contrast to standard Morse theory one cannot estimate the dimension of thehomology of the sublevels of distance functions (i.e. of metric balls) directly since theintersection of a ball with the cutlocus can be rather complicated. As a replacementfor this part of the Morse theory Gromov introduces the concept of content:De�nition 3.20 Let Y � X be open subsets of M . The content of Y in X is de�nedas the rank of the inclusion map on the homology levelcont(Y;X) := rk(H�(Y )! H�(X)) :The content of a metric ball B in M is de�ned ascont(B) := cont(B; 5B) :The content of B is a measure for how much of its homology survives after the inclusionmap into 5B . Clearly cont(B) = 1 for any contractible ball B . By corollary 3.17 andthe Isotopy Lemma 3.12 for su�ciently large balls B there is an isotopy of M whichmoves M into B . Therfore there is a map f :M !M such that the induced map f�is the identity on H�(M) and f(M) � B � 5B �M . Hence cont(B) = cont(M;M) =dimF H�(M).The strategy for the proof now consists in showing that the content of any metricball and hence of M is bounded by a constant C(n). For this purpose Gromov in-troduces the concepts of corank and compressibility for metric balls with the followingproperties:(i) Either a ball of content > 1 is incompressible or it can be deformed into anincompressible smaller ball of at least the same content and of at least the samecorank.(ii) The corank is bounded by a constant k0 � 2n.35

(iii) If a ball B of radius r and of corank k is incompressible, then any ball of radius� r5L with center in 32B has corank at least k + 1.(iv) A ball with maximal corank has content 1.Now the proof is based on a reverse induction over the corank: By (i) only incom-pressible balls need to be considered. Suppose that the content of any ball of corank> k is bounded by ak(n). Let B be an incompressible ball of radius r and corank k .Then B is covered by balls Bi of radius � = r5L�10n+1 such that the concentric balls12Bi are disjoint. The maximal number N of these balls can be estimated from aboveby the Bishop-Gromov volume comparison argument. It depends only on n. Usingproperty (iii) and the induction assumption, a topological argument stemming from ageneralized Mayer-Vietoris sequence for nested coverings then is used to show that thecontent of B is bounded by ak(n) �Nn+1 , completing the induction argument.We start introducing the concept of compressibility which essentially correspondsto "�� compressibility" used by Abresch with the �xed value � = 5.De�nition 3.21 A ball B of radius r in M is called compressible if there is a ball ~Bof radius ~r � 35r around some point in 2B such that there is an isotopy of M whichis �xed outside 5B and which moves B into ~B . Briey we say that B is compressibleinto ~B when these conditions hold.If B is compressible into ~B , then ~B � 5 ~B � 5B and the pairs (5B;B) and (5B; ~B)are homotopically equivalent. Therefore it is clear thatcont(B) � cont( ~B) :Consequently for each ball B of content > 1 there is an incompressible ball B0 �5B0 � 5B such that cont(B0) � cont(B). For this observe that the injectivity radiuson the compact ball 5B is bounded below by some constant � > 0 and if a ball canbe compressed successively into a �nal ball of radius � � then it must have content 1since the � -balls are contractible.Lemma 3.22 Suppose B is an incompressible ball of radius r . Then for any point~p22B there must be a critical point ~q for the distance function dist~p in the compactannulus A~p = B(~p; 3r) nB(~p; 35r).Proof. Suppose for some point ~p22B there is no critical point in A~p . Let ~B =B(~p; 35r). Then we have inclusions B � B(~p; 3r) and B(~p; 3r) � 5B . By the isotopylemma 3.12 there is an isotopy of M which is �xed outside 5B moving B(~p; 3r) and36

hence B into ~B contradicting the incompressibility of B . 2De�nition 3.23 Given p2M , r > 0. Let kr(p) be the maximal number of criticalpoints qj , j = 1; : : : ; kr(p), for distp satisfyingdistp(qj) � 3Lr and distp(qj+1) � 1Ldistp(qj) :The corank of the ball B = B(p; r) is de�ned ascorank(B) = inffkr(~p) j ~p25Bg :Note that kr(p) � 2n and therefore corank(B) � 2n by the choice of L. If B iscompressible into ~B , then corank(B) � corank( ~B).As an immediate consequence of the previous Lemma 3.22 we haveCorollary 3.24 Suppose B = B(p; r) is incompressible and r̂ � r5L .a) If ~p22B , then kr̂(~p) � 1 + corank(B).b) If p̂2 32B and B̂ = B(p̂; r̂), then corank(B̂) � 1 + corank(B).Proof. For a) let qj , 1 � j � kr(~p) =: k be critical points with dist~p(qj) � 3Lr anddist~p(qj+1) � 1Ldist~p(qj). Since B is incompressible, there is a critical point qk+1 for ~pin the annulus A~p as in lemma 3.22. Thus 3Lr̂ � 35r � dist~p(qk+1) � 3r � 1Ldist~p(qj).Now the kr(~p) + 1 points qj satisfy the condition for the de�nition of kr̂(~p) hence a)follows.For b) observe the inclusions 5B̂ � (32 + 1L)B � 2B . Now kr̂(~p) � 1 + corank(B) forany ~p25B̂ . 2As a consequence a ball of maximal corank must have content 1: If B has maximalcorank, then because of b) in the lemma, B must be compressible into a ball of radius35r with the same maximal corank and at least the same content. This procedure canbe repeated k times until one reaches a ball ~B of radius (35)kr which is smaller than theinjectivity radius on the compact ball 5B . Then cont( ~B) = 1 and hence cont(B) = 1.These are the basic ingredients from critical point theory. We now turn to thecovering arguments.Lemma 3.25 Given an n-dim Riemannian manifold M of nonnegative Ricci curva-ture, a ball B of radius r and a covering of B by balls B1; : : : ; BN of radius " � rwith center in B such that the corresponding balls 12B1; : : : ; 12BN are disjoint. ThenN � (6r")n :37

Proof. Choose i0 such that the ball 12Bi0 with center p0 has the smallest volumeamong all the given balls. The ball B̂ around p0 of radius 3r contains all the Bi .Therefore N � volB̂vol12Bi0 � 3r12"!nwhere the last inequality is the Bishop-Gromov estimate for the volume of concentricballs, which has been discussed in the �rst series of these lectures given by K. Grove,compare also [K] for a proof. 2Since the proof of theorem 3.19 will be based on reverse induction over the corank,we introduce the following notation:Let k0 � 2n be the maximal corank of metric balls. For 0 � k � k0 we denote by Bkthe set of balls having corank � k .The topological information for the induction step is contained in the next lemma:Lemma 3.26 Suppose cont( ~B) is bounded by a constant ak for any ~B2Bk . Further-more let B2Bk�1 be incompressible. Thencont(B) � ak �Nn+1where N � (3L � 10n+2)n .Proof. Choose a covering of B by balls B1; : : : ; BN of radius "(r) = r5L�10n+1 such that12B1; : : : ; 12BN are disjoint. Then by lemma 3.25 N � (3L � 10n+2)n . For 0 � j � n+1we also consider the coverings Bj1; : : : ; BjN where Bji = 10j � Bi . The radii of allthese balls are � r5L . By corollary 3.24 we have corank(Bji ) � 1 + corank(B) � k ,hence cont(Bji ) � ak . Using the result on the nested coverings in corollary 4.2 of theappendix, we obtaincont([i B 0i ;[i B n+1i ) � nX̀=0 Xi0

The �rst chain of inclusions implies cont(B) � cont(Si B 0i ;Si B n+1i ), and fromthe second we conclude that the content of any of the intersections is bounded bycont(Bn�`i0 ) � ak . Since the number of terms in the sum on the right hand side isbounded by Nn+1 , compare (81) in the appendix, the proof is complete. 2Proof of Theorem 3.19 a): Reverse induction over the corank: For B2Bk0 we havecont(B) = 1. Assume now that cont(B) � ak(n) for any B2Bk . Let B2Bk�1 If Bis compressible and cont(B) > 1, then B can be compressed into a ball of at leastthe same content and of at least the same corank. Therefore we can assume that Bis incompressible. Now lemma 3.26 applies and we get cont(B) � ak(n) �Nn+1 . Sincek0 � 2n we get recursivelydimF H�(M) = cont(M) � N2n2+2nwhere N = (3L � 10n+2)n , L = 3(1 + p2)n�1 . Using L < 3n+1 , an explicit roughestimate for C(n) is given by C(n) � 103n4+9n3+6n2 : 2Remarks1. Note that the exponent in the estimate for C(n) is a polynomial of order 4 in n.Gromov's original constant depended double exponentially on n. The reason forthis improvement due to Abresch is the choice of L, the modi�cation of corankand compressibility to eliminate one of Gromov's critical point lemmas whichall together gave a better estimate for the corank, and �nally the improvementof the estimate in the inductive lemma 3.26 where Gromov uses the estimatecont(B) � ak � 2N .2. The estimate for the constant C(n) still seems to be far away from reality. Knownexamples of n-manifolds with nonnegative curvature all have a sum of Betti num-bers � 2n .3.6 The soul theoremThis �nal section is devoted to the soul theorem, cf. [GM1], [CG1].Theorem 3.27 (Cheeger, Gromoll) Let Mn be a complete noncompact manifold ofnonnegative curvature K . Then there is a compact totally geodesic submanifold S in39

M such that M is di�eomorphic to the normal bundle �(S) of S . If K > 0, then Mis di�eomorphic to IRn .We �rst introduce a few basic concepts which are needed for the proof.De�nition 3.28 A nonempty subset C of M is called totally convex if for arbitrarypoints p; q2C any geodesic with endpoints p and q is contained in C .De�nition 3.29 A ray in M is a normal geodesic c : [0;1[!M for which any �nitesegment is minimal. For a ray c : [0;1[!M we de�ne the halfspaces Bc respectivelyHc by Bc = [t>0B(c(t); t)Hc = M nBcwhere B(c(t); t) is the open metric ball of radius t around c(t).Note that in a complete noncompact manifold M for any p2M there exists a rayc : [0;1[! M with initial point c(0) = p. For a sequence qi2M with limi!1(p; qi) =1 and normal minimal geodesics ci from p to qi any limiting geodesic c obtainedfrom a convergent subsequence of ci will be a ray emanating from p. ( _ci(0) has anaccumulation point in the compact unit sphere in TpM ).The basic observation about the halfspaces Hc is the following.Lemma 3.30 If M is complete, noncompact of nonnegative sectional curvature, thenHc is totally convex for any ray in M .Proof. Suppose Hc is not totally convex, i.e. there is a geodesic c0 : [0; 1]!M withendpoints c0(0); c0(1)2Hc but c0(s)2Bc for some s2 ]0; 1[. Then q := c0(s)2B(c(t0); t0)for some t0 > 0 and hence q2B(c(t); t) for any t � t0 by the triangle inequality: Infact setting t0 � " = dist(q; c(t0)); " > 0we have dist(q; c(t)) � dist(q; c(t0)) + dist(c(to); c(t))= (t0 � ") + (t� t0) = t� "for t � t0 . 40

Let c0(st) be a point on c0 which is closest to c(t). Further consider the restrictionct0 := (c0j[0;st])�1 and a minimal geodesic ct1 from c0(st) to c(t). Since ct0(0) = c0(st)is the closest point to c(t) on c0 we have

p

Ct

Figure 6: paraboloidThe Ct provide an expanding �ltration of M by compact totally convex sets. Our nextgoal is to construct minimal totally convex sets by a contraction procedure which willbe used to �nd a soul S . For this important part of the proof we also need the localconcept of convexity:De�nition 3.32 A subset A of M is called strongly convex if for any q; q0 2A thereis a unique minimal geodesic from q to q0 which is contained in A.Recall that there is a continuous function r :M !]0;1], the convexity radius suchthat for any p2M , any open metric ball B which is contained in B(p; r(p)) is stronglyconvex, cf [GKM].De�nition 3.33 We say that a subset C of M is convex if for any p2C there is anumber 0 < "(p) < r(p) such that C \ B(p; "(p)) is strongly convex.Note that a totally convex set is convex and connected. Also the closure of a convexset is again convex.Let C be a connected nonempty convex subset of M . For 0 � l � n we mayconsider the collection fN l�g of smooth l�dim submanifolds of M such that N l� � C .Let k denote the largest integer such that fNk�g is nonempty and N := S�Nk� � C .Lemma 3.34 N is a smooth connected totally geodesic submanifold of M and C � N .Moreover N = C is a topological manifold with possibly empty bounary @N = N nN .Proof (outline). The full details are technical, therefore we only give the mainidea, [CG1], [CE]. Let p2N and "(p) as in the de�nition above. Then p2Nk� forsome � . Therefore we can choose a neighborhood U � N� \ B(p; 12"(p)) of p in42

N and 0 < � < 12"(p) such that exp j��(U) is a di�eomorphism onto a neighborhoodT� of p in M , where ��(U) = fv2(TU)? j kvk < �g � TM is the � -tube in thenormal bundle �(U) of U . To prove that N is a submanifold it su�ces to showthat N \ T� = U . Suppose q2(N \ T�) n U � (C \ T�) n U . Let q0 be the closestpoint to q in U . Then q0 2U , otherwise we get a contradiction to the invertibilityof exp j��(U) close to q . The minimal geodesic from q to q0 then is orthogonal to U .By the choice of � < 12"(p) the exponential map in the ball of radius � around q0 isinvertible. Therefore all the unique minimal geodesics from q to q00 for q00 in someneighborhood U 0 of q0 are transversal to U and are contained in C . The conical setfexp tu j u2Mq; kuk < "(q); exp(u)2U 0; 0 < t < 1g then is a (k + 1)-dimensionalsubmanifold in C which contradicts the de�nition of k . From the existence of T� andthe convexity of C it follows that N is totally geodesic. For the remaining statementswe refer to [CG1] and [CE]. 2De�nition 3.35 Let C be a convex subset of M . The tangent cone to C at a pointp2C is by de�nition the setTpC = fv2TpM j exp(t vkvk)2N for some 0 < t < r(p)g [ f0g :Clearly if p2N = int (C), then TpC = TpN . The following lemma contains all thetechnical information about TpC we need.Lemma 3.36 (tangent cone lemma) Let C �M be convex and p2@C .a) Then TpC n f0g is contained in an open halfspace of TpM .b) Suppose that there exists q2intC and a minimal normal geodesic c : [0; d] ! Cfrom q to p such that jcj = dist(q; @C). ThenTpC n f0g = fv2 T̂pC j

Lemma 3.37 (contraction lemma) Suppose M has nonnegative sectional curva-ture and C �M is a closed totally convex subset with @C 6= ;. We setCa = fp2C j dist(p; @C) � ag ; Cmax = \Ca 6=;Ca :Thena) Ca is closed and totally convex.b) dimCmax < dimC .c) If K > 0 then Cmax is a point.This is a corollary of the following more general lemma:Lemma 3.38 Under the assumptions of lemma 3.37, := dist@C : M ! IR is aconcave function, i.e. for any normal geodesic c which is contained in C we have (c(�t1 + (1� �)t2)) � � (c(t1)) + (1� �) (c(t2)) : (65)If the sectional cuvature satis�es K > 0 then the strict inequality holds in (65).Proof. It is su�cient to show that for any point c(s0) of c there is a number � > 0such that (c(s)) is bounded above by a linear function h(s) on ]s0��; s0+�[ satisfyingh(s0) = (c(s0)) =: d. Let cs0 be a distance minimizing normal geodesic of length dfrom c(s0) to @C and � := �2 , � < �2 . Note thatwe only have to consider points s � s0 .Case � = �2 : Let E denote the parallel unit vector �eld along cs0 with E(0) = _c(s0).By the second comparison theorem of Rauch, there is a number � > 0 such that thelength of the curve cs(t) = exp(s�s0)E(t) has length jcsj � d = jcs0j for 0 � s�s0 � � .The geodesic �c : s 7! exp(s � s0)E(d) is orthogonal to cs0 at q := cs0(d)2@C , hence_�c(0) 62TqC by lemma 3.36, so that �c(t) 62 intC for 0 < t < "(q). Therefore (c(s)) �jcsj � d = d� (s� s0) cos �2 .Case � > �2 : Let E(0)? _cs0(0) be the unique unit vector in the convex cone spannedby _c(s0) and _cs0(0) and extend it to the parallel vector �eld E along cs0 . De�ne cs asin the �rst case to obtain jcsj � d : (66)44

Applying the hinge version of Toponogov's theorem (or just Rauch I) to the hinge withgeodesics t 7! exp tE(0), 0 � t � (s � s0) cos(� � �2 ) and t 7! c(s0 + t) with angle�� �2 , one obtainsdist�c(s); exp((s� s0) cos(�� �2 )E(0)� � �(s� s0) cos� : (67)Combining (66) and (67), the inequality (c(s)) � d� (s� s0) cos� follows.Case � < �2 : Choose the point cs0(ts) on cs0 such that dist(c(s); cs0([0; d])) =dist(c(s); cs0(ts)) and a normal minimal geodesic as from cs0(ts) to c(s).Then

such initial vectors are contained in an open half space of TqM , compare lemma 3.36.Therefore distS has no critical points on M nS . Choose " > 0 such that expj�"(S) is adi�eomorphism onto the "-tube around S . Here �"(S) = fv2TS? j kvk < "g. ThenX1 = grad distS is a gradient-like vector �eld on exp(�"(S))nS such that hX1jq; _cq(0)i= �1 for the unique minimal normal geodesic cq from q to S . Therefore one canconstruct a global gradient-like vector �eld X on M n S such that hXq; _c(0)i < 0 forany distance minimizing geodesic from q to S and Xq = X1jq for q2 exp(�"=2(S)). Let be the ow of X . De�ne F : �(S) ! M as follows: F (v) := exp(v) for kvk � "4and F (tv) := (exp( "4v); t� "4) for v2�1(S) and t � "4 . Then F is a di�eomorphismas follows easily by using (64). 2Remarks1. A soul of M is not uniquely determined in general as can be seen by looking atcylinders. However any two souls of M are isometric, cf. [S] and [Y].2. If codim(S) = 1, then exp j�(S) is an is an isometry between �(S) with itsstandard (at) bundle metric and M , cf. [CG1].3. In general the normal bundle �(S) need not to be trivial. Furthermore M is notlocally isometric to a product S � IRk in general. By the Toponogov splittingtheorem, cf. [CG1], however any line in M splits o� isometrically, so that M isisometric to �M� IRk , where IRk carries the standard at metric and �M does notcontain any lines. This even holds for manifolds of nonnegative Ricci curvature,cf. [CG2], [EH]. More generally Strake [St] has shown the following: Supposethe holonomy group of �(S) is trivial, then M is isometric to S� IRk where IRkcarries a metric of nonnegative curvature. For further results in this context wealso refer to [ESS].4. For a discussion on the structure of the fundamental group see [CG1].5. There is no analogue of the soul theorem for complete open manifolds of positiveRicci curvature, cf. the examples in [GM2], [SY1] and [B], but compare also theresult in [AG].46

4 Appendix: A topological LemmaTheorem 4.1 (Nested Coverings) Let B 0i � B 1i � : : : � Bm+1i ,1 � i � N , be afamily of nested open subsets in a topological space X , and let X j := SNi=1 B ji for0 � j � m + 1. Thenrk�Hp(X0)! Hp(Xp+1)�� pXk=0 Xi0

Taking the corresponding long exact homology sequences leads in particular to boththe commutative diagrams with exact rows:Hp(C0�;q) - Hp(A0�;q) - Hp�1(A0�;q+1)? ?��0;pp;q ?��0;pp�1;q+1Hp(Cp�;q) - Hp(Ap�;q) - Hp�1(Ap�;q+1)?�p;p+1p;q ?��p;p+1p;q ?Hp(Cp+1�;q ) - Hp(Ap+1�;q ) - Hp�1(Ap+1�;q+1)(75)

when 1 � p � m, andH0(C0�;q) - H0(A0�;q) - H�1(A0�;q+1) = 0?�0;10;q ?��0;10;q ?H0(C1�;q) - H0(A1�;q) - H�1(A1�;q+1) = 0 (76)else. Here the vanishing occurs already on the chain level: Aj�1;q+1 � Cj�1;q = 0.When applying standard diagram chasing techniques, (75) and (76) yield the followingestimates respectively:rk(��0;p+1p;q ) = rk(��p;p+1p;q � ��0;pp;q)� rk(�p;p+1p;q ) + rk(��0;pp�1;q+1) for 1 � p � m (77)rk (��0;10;q) � rk(�0;10;q ) (78)By induction we conclude thatrk(��0;p+1p;q ) � pXk=0 rk(�k;k+1k;p+q�k) = pXk=0 rk(�p�k;p+1�kp�k;q+k ) (79)for 0 � p � m. Setting q to 0, this inequality specializes | in the presence of formulae(70), (72) , and (73) | precisely to the claim in Theorem 4.1. 2.Corollary 4.2 (Nested Coverings) Let B 0i � B 1i � : : : � B n+1i ,1 � i � N , be afamily of nested open subsets in an n-dimensional topological manifold Mn . Thenrk H�([i B0i )! H�([i Bn+1i )!� nXk=0 Xi0

Remark: The number of terms on the r.h.s. of (80) isnXk=00@ Nk + 11A < N � nXk=0 Nk(k + 1)! < Nn+1 (81)Proof of the Corollary. Since we are dealing with open subsets in an n-dimensionalmanifold Mn , Hp vanishes unless 0 � p � n. Thereforerk H�([i B0i )! H�([i Bn+1i )!= nXp=0 rk Hp([i B0i )! Hp([i Bn+1i )!� nXp=0 rk Hp([i Bn�pi )! Hp([i Bn+1i )!Each term on the r.h.s. can be estimated separately by applying Theorem 4.1 to thenested open sets B n�pi � : : : � B n+1i , 1 � i � N . With this shift in the indexing inmind [ m + 1 = (n+ 1)� (n� p) ], one gets { slightly sharper than (80) { :rk H�([i B0i )! H�([i Bn+1i )!� nXk=0 Xi0

References[A] Abresch, U. : Lower Curvature Bounds, Toponogov's Theorem, and BoundedTopology I/II. Ann. scient. �Ec. Norm. Sup., 4e s�erie, t. 18, 1985, 651{670.[AG] Abresch, U., Gromoll, D. : Open manifolds with nonnegative Ricci curvature.[AW] Alo�, S., Wallach, N. R. : An in�nite family of distinct 7-manifolds admittingpositively curved Riemannian structures. Bull. Amer. Math. Soc. 81 (1975),93-97.[B] B�erard Bergery, L. : Quelques exemples de vari�et�es riemanniennes compl�etesnon compactes �a courbure de Ricci positive. C. R. Acad. Sc. Paris, t. 302. S�erieI, no 4, (1986), 159-161.[BT] Bott, R., Tu, L. W. : Di�erential Forms in Algebraic Topology. Graduate Textsin Mathematics 82, Springer 1982.[CE] Cheeger, J., Ebin, D.G. : Comparison Theorems in Riemannian Geometry.American Elsevier, New York 1975.[CG1] Cheeger, J., Gromoll, D. : On the structure of complete manifolds of nonneg-ative curvature. Ann. of Math. 96 (1972), 413-443.[CG2] Cheeger, J., Gromoll, D. : The splitting theorem for manifolds of nonnegativeRicci curvature. J. Di�. Geom. 6 (1971), 119-128.[Cg] Cheng, S. Y. : Eigenvalue comparison theorem and its geometric applications.Math. Z. 143 (1975), 289-297.[E] Elerath, D.: An Improved Toponogov Comparison Theorem for Non-negativlyCurved Manifolds. J. Di�. Geom. 15 (1980), 187-216.[EH] Eschenburg, J., Heintze, E. : An elementary proof of the Cheeger-Gromollsplitting theorem. Ann. Glob. Analysis and Geometry 2 (1984), 141-151.[ESS] Eschenburg, J., Schroeder, V., Strake, M. : Curvature at in�nity of opennonnegatively curved manifolds. J. Di�. Geom. 30 (1989), 155-166.[GG] Gromoll, D., Grove, K. : Rigidity of positively curved manifolds with largediameter. Seminar on Di�erential Geometry, Ann. of Math. Studies, PrincetonUniversity Press (1982), 203-207. 50

[GKM] Gromoll, D., Klingenberg, W., Meyer,W. : Riemannsche Geometrie imGrossen. Lecture Notes 55, Springer 1968.[GM1] Gromoll, D., Meyer, W. : On complete manifolds of positive curvature. Ann.of Math. 90 (1969), 75-90.[GM2] Gromoll, D., Meyer, W. : Examples of complete manifolds with positive Riccicurvature. J. Di�. Geom. 21 (1985), 195-211.[G1] Gromov, M. : Almost at manifolds. J. Di�. Geom. 13(2) (1978), 231-243.[G2] Gromov, M. : Curvature, diameter and Betti numbers. Comment. Math. Helv.56 (1981), 179-195.[GS] Grove, K., Shiohama, A. : A generalized sphere theorem. Ann. of Math. 106(1977), 201-211.[GP] Grove, K., Petersen, P. : Bounding homotopy types by geometry. Ann. ofMath. 128 (1988), 195-206.[K] Karcher, H. : Riemannian Comparison Constructions. Preprint Bonn 1987[SY1] Sha, Ji-Ping, Yang, Da-Gang : Examples of manifolds of positive Ricci curva-ture. J. Di�. Geo. (1989)[SY2] Sha, Ji-Ping, Yang, Da-Gang : Positive Ricci Curvature on the ConnectedSums of Sn � Sm . Preprint.[S] Sharafutdinov, V. A. : Convex sets in a manifold of negative curvature. Math.Zametki 26 (1979) 556-560.[St] Strake, M. : A splitting theorem for open nonnegatively curved manifolds.Manuscripta Math. 61 (1988) 315-325.[W] Wallach, N. L. : Compact homogeneous Riemannian manifolds with strictlypositive curvature. Ann. of Math. 96 (1972) 277-295.[Y] Yim, J. W. : Distance nondecreasing retraction on a complete open manifoldof nonnegative curvature. Preprint, University of Pennsylvania, 1987.51