Abstract We prove an analogue of the Cheeger–Gromoll splitting theorem for sub-Riemannian manifolds with the measure contraction property instead of the nonnegativity ofthe Ricci curvature. If such a sub-Riemannian manifold contains a straight line, then the man-ifold splits diffeomorphically, where the splitting is not necessarily isometric. We prove thatsuch a sub-Riemannian manifold containing a straight line cannot split isometrically undersome typical condition in sub-Riemannian geometry. Heisenberg groups are such examples.
The celebrated splitting theorem of Cheeger and Gromoll is a very important theorem inRiemannian geometry. It has many applications to study the topology of a compact Rie-mannian manifold with nonnegative Ricci curvature [6,7] .
Theorem 1 (Cheeger and Gromoll [7]) Let (M, g) be a complete Riemannian manifoldwith nonnegative Ricci curvature. If (M, g) contains a straight line γ , then there exists aRiemannian manifold X such that M is isometric to the Riemannian product X × R.
We provide some sub-Riemannian versions of Theorem 1. Let M be a connectedn-dimensional differentiable manifold and D ⊂ T M a subbundle, where Dp for any pointp ∈ M is endowed with an inner product 〈·, ·〉p . We assume that D satisfies Hörmander’scondition (see Definition 1) and is regular at any point p ∈ M [24, Definition 2.8]. Let dbe the sub-Riemannian distance function associated with D and 〈·, ·〉. We give a Lebesguemeasure μ on M , i.e., a measure that is absolutely continuous with respect to the Lebesgue
K. Itoh (B)Mathematical Institute, Tohoku University, Sendai 980-8578, Japane-mail: [email protected]
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measure on each chart and has continuous density. We consider the following condition forthe triple (M, d, μ).
(*) (M,D, d) is a complete sub-Riemannian manifold that does not contain nontrivial sin-gular minimizers. (M, d, μ) satisfies MCP(0, N ) for a real number N ∈ [1,+∞).
We explain the notion of a singular minimizer in Definition 2. MCP(0, N ) is the measurecontraction property explained in Definition 6. In Theorem 1, we replace a Riemannianmanifold by a sub-Riemannian manifold and the nonnegative Ricci curvature condition byMCP(0, N ), and prove the following main theorem.
Theorem 2 If (M, d, μ) satisfies the condition (∗) and contains a straight line γ , then thefollowing (1), (2) and (3) hold.
(1) The Busemann functionbγ of γ is smooth.(2) Any integral curve of ∇bγ is a straight line.(3) The 1-parameter group of ∇bγ gives a diffeomorphism between M and b−1
γ (0)× R.
We recall the notion of a straight line and a Busemann function in Sect. 5. To state anothermain theorem, we consider the following condition for the subbundle D of T M .
(**) There exists a point p ∈ M such that for any vector v ∈ Dp − {0}, there exists a vectorw ∈ Dp satisfying [V,W ]p /∈ D p , where V,W are any extensions of v,w to vectorfields tangent to D respectively.
Theorem 3 If (M, d, μ) satisfies the condition (∗) and if D satisfies the condition (∗∗), thenthere does not exist any metric space X such that M is isometric to the product metric spaceX × R.
We prove analytic tools like the Laplacian comparison theorem and the maximum prin-ciple to obtain Theorems 2 and 3. The measure contraction property is a suitable notion forTheorems 2 and 3 providing those tools. This is a generalization of the condition that theRicci curvature is bounded from below by a constant κ ∈ R. For example, the mean cur-vature comparison theorem of metric spheres (see [5, Appendix 2] and [14, Chapter 5.I+])and studies of the heat kernel and the diffusion process associated with a Dirichlet form ona metric measure space by Sturm [34] gave some motivations studying the measure contrac-tion property. For an n-dimensional Riemannian manifold, MCP(κ, n) is equivalent to thecondition Ric ≥ κ [26]. There are some geometric studies for metric measure spaces withMCP(κ, N ) [26,27,37].
We obtain sub-Riemannian manifolds with MCP(0, N ) that split diffeomorphically butdo not split isometrically, from Theorems 2 and 3. For example, the (2n + 1)-dimensionalHeisenberg group satisfies the assumptions of Theorems 2 and 3. Since there are no suchexamples in Riemannian geometry, our main theorems show that the metric structure of sub-Riemannian geometry is more complicated than the Riemannian one. If we do not assume thecondition (∗∗), then there is an example of sub-Riemannian manifolds splitting isometrically.See Remark 6. Kuwae and Shioya [22] studied a splitting theorem for Alexandrov spaces,where an Alexandrov space is a metric space generalizing the notion of a Riemannian man-ifold with sectional curvature bounded from below. They proved that a splitting theorem forAlexandrov spaces holds homeomorphically under the measure contraction property, how-ever it is not known whether or not it holds isometrically. R
n with a norm and the Lebesguemeasure satisfies MCP(0, n) and contains a straight line but does not split isometrically ingeneral.
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Agrachev and Lee [1] proved that for a contact distribution on a 3-dimensional manifold,the nonnegativity of the generalized Ricci curvature defined in implies MCP(0, 5) for anappropriate measure. Combining Theorem 2 with this, we have the following corollary.
Corollary 1 Let (M,D, d) be a compact 3-dimensional contact sub-Riemannian manifoldwith nonnegative generalized Ricci curvature in the sense of [1]. If the fundamental groupπ1(M) of M is an infinite group, then the universal covering space M of M is diffeomorphicto either S2 × R or R
3.
A rough sketch of the proof of Theorem 2 comes from that of Cheeger and Gromoll [7].The main differences are the proofs of the Laplacian comparison theorem and a maximumprinciple from MCP(κ, N ). The Laplacian comparison theorem is proved according to anidea of [22]. Our maximum principle is induced from Kuwae’s maximum principle [21]that is stated in the framework of Dirichlet forms. In our setting, we define a Dirichlet formassociated with the sub-Riemannian structure. We use analyses in [15,28,33] to check theassumptions in Kuwae’s maximum principle.
The condition (∗∗) means that the subbundle D is twisted at a point for every directionof D. In the proof of Theorem 3, by contradiction, we assume that there exists a metricspace X such that the sub-Riemannian manifold M is isometric to X × R. Then the naturalidentification map between any two sections X × {t1} and X × {t2} is an isometry. Howeverthis contradicts the condition (∗∗).
Remark 1 (1) There is another condition that the Ricci curvature is bounded from belowfor general metric measure spaces, the curvature dimension condition CD(κ, N ) (κ ∈R, N ∈ [1,+∞]), defined in [23,35,36]. CD(κ, N ) implies MCP(κ, N ) in our setting[36, Theorem 5.4]. The (2n + 1)-dimensional Heisenberg group with the Lebesguemeasure does not satisfy CD(κ, N ) for any κ ∈ R nor any N ∈ [1,+∞], but satisfiesMCP(0, 2n + 3) [20]. Moreover, it is not an Alexandrov space [20].
(2) Zhang and Zhu [38] proved an isometric splitting theorem for Alexandrov spaces undera condition that the Ricci curvature is bounded from below stronger than MCP(κ, N )and CD(κ, N ).
(3) If the subbundle D is a fat distribution (see Definition 3) and D = T M , then D satisfiesthe condition (∗∗). In particular, every contact distribution satisfies the condition (∗∗).
(4) A sub-Riemannian manifold with nonnegative generalized Ricci curvature in the sense of[1] with the appropriate measure satisfies the assumptions of Theorem 3, since any sub-Riemannian manifold associated with a contact distribution does not contain nontrivialsingular minimizers. For example, the 3-dimensional Heisenberg group is such a sub-Riemannian manifold.
The organization of this paper is as follows. In Sect. 2, we mention some fundamental def-initions and results of geodesics on a sub-Riemannian manifold and the measure contractionproperty. In Sect. 3, we prove the Laplacian comparison theorem. In Sect. 4, we present theDirichlet form associated with the sub-Riemannian structure and prove a maximum principle(Proposition 2) by using Kuwae’s [21] maximum principle . In Sect. 5, we prove Theorems2, 3 and Corollary 1. In Sect. 6, we give an example of sub-Riemannian manifolds satisfyingthe assumptions of Theorems 2 and 3.
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2 Preliminaries
We define a distance between any two points p, q ∈ M , by the infimum of the length of allabsolutely continuous curves c : [0, 1] −→ M [24, §1.4] satisfying c(t) ∈ Dc(t) for almostevery t ∈ [0, 1] and c(0) = p, c(1) = q ,
d(p, q) := infc
1∫
0
|c(t)|dt.
Note that, in general, the topology given by d(·, ·) does not coincide with the topology of themanifold M .
Definition 1 (see [17,24]) Let m ∈ N be the rank of the subbundle D. We say that D satisfiesHörmander’s condition if there exists a number r ∈ N such that for any point p ∈ M andany local frame X1, . . . , Xm of D around p, iterated Lie brackets of X1, . . . , Xm ,
X1, . . . , Xm, [Xi , X j ], . . . , [Xi1 , [Xi2 , [. . . , [Xir−1 , Xir ] . . . ]]]for all i, j, . . . , i1, . . . , ir = 1, . . . ,m span the tangent space Tp M .
Theorem 4 (Chow–Rashevskii Theorem) If the subbundle D ⊂ T M satisfies Hörmander’scondition, then the distance function d(·, ·) attains finite value and gives the topology of themanifold M.
In this paper we call the distance function d a sub-Riemannian distance function andthe triple (M,D, d) a sub-Riemannian manifold if the subbundle D satisfies Hörmander’scondition.
Next we mention some basic notions about sub-Riemannian geodesics. We say that anabsolutely continuous curve γ : [0, T ] −→ M, T > 0, is a minimizing geodesic if γsatisfies γ (t) ∈ Dc(t) for almost all t ∈ [0, T ] and
∫ T0 |γ | dt = d(γ (0), γ (T )). We call γ
a geodesic if γ is a locally minimizing geodesic. For any point p0 ∈ M and any positivenumber T > 0 we denote by Ω([0, T ], p0 ; D) the set of all absolutely continuous curvesc : [0, T ] −→ M that start at p0 and satisfy
∫ T0 |c(t)|2 dt < +∞. We define the endpoint
map end: Ω([0, T ], p0 ; D) −→ M by end(c) := c(T ) for any curve c ∈ Ω([0, T ], p0 ; D).We have a suitable differential structure on Ω([0, T ], p0 ; D) so that the endpoint map isdifferentiable [24].
Definition 2 A singular curve is a critical point of the endpoint map. If a singular curve isa geodesic, we call it a singular geodesic, and if it is a minimizing geodesic, we call it asingular minimizer.
In sub-Riemannian (non-Riemannian) manifolds, all constant curves are singular curves.We call a nonconstant singular curve a nontrivial singular curve. The behavior of singularcurves is very complicated and much is unknown in general. In this paper we assume thatthe sub-Riemannian manifold (M,D, d) does not contain nontrivial singular minimizers.There is a famous class of sub-Riemannian manifolds that do not contain nontrivial singularminimizers.
Definition 3 We call D a fat distribution or strong bracket generating distribution if thefollowing holds. Let p be any point of M and X1, . . . , Xm a local frame of D around p. Forany section X of D that does not vanish at p, the subspace spanned by the values at p of
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X1, . . . , Xm, [X1, X ], . . . , [Xm, X ]coincides with the tangent space Tp M .
If D is a fat distribution, then (M,D, d) does not contain nontrivial singular minimizers[24]. Any contact distribution is a fat distribution [24]. According to [8,9,11], there is anothermethod to obtain classes of sub-Riemannian manifolds that do not contain nontrivial singularminimizers.
In general we cannot write the sub-Riemannian geodesic equation as second order ODE ofthe coordinate of the curve (x1(t), . . . , xn(t)) such as the Riemannian case. We consider theHamiltonian equation on the cotangent bundle T ∗M and the exponential maps by definingthe Hamiltonian function adapted to the sub-Riemannian strucure in the following [24].
We define a linear map gp : T ∗p M −→ Dp by 〈 gp(ξ), v 〉p := ξ(v) for any p ∈
M, v ∈ Dp, ξ ∈ T ∗p M . We lift the metric 〈·, ·〉p to the cotangent space T ∗
p M as (ξ, η)p :=〈 gp(ξ), gp(η) 〉p for any ξ, η ∈ T ∗
p M . We define a function on the cotangent bundle T ∗Mby H(ξ) := (ξ, ξ)p/2 for any p ∈ M and ξ ∈ T ∗
p M . Then the Hamiltonian vector fieldH is determined by using the canonical symplectic structure of T ∗M regarding H as theHamiltonian function. The Hamiltonian equation is written as ξ (t) = Hξ(t). If we take anyinitial value ξ0 ∈ T ∗
p M , then the solution of this equation exists and is uniquely determined.We can define the exponential map expp : T ∗
p M −→ M for each p ∈ M by expp(ξ0) :=π ◦ ξ(1) for any ξ0 ∈ T ∗
p M , where π : T ∗M −→ M is the canonical projection of thecotangent bundle T ∗M and ξ : [0, 1] −→ T ∗M is the unique solution of the Hamiltonianequation with ξ0 as the initial value. The curve π ◦ ξ(t) is always a geodesic on M [24].
Definition 4 A curve c : [0, 1] −→ M is a normal geodesic if there exists a solutionξ : [0, 1] −→ T ∗M of the Hamiltonian equation such that c = π ◦ ξ . We call ξ a lift of c onthe cotangent bundle T ∗M .
Remark 2 In general there exist sub-Riemannian geodesics that do not satisfy the Hamil-tonian equation. Such geodesics behave quite different from the Riemannian ones, and itis not known whether they are always smooth or not [24]. Since we have assumed thatthe sub-Riemannian manifold (M,D, d) does not admit nontrivial singular minimizers, allminimizing geodesics on M are normal [24].
Next we mention the cut locus of a point in a sub-Riemannian manifold.
Definition 5 For a point p ∈ M , a point q ∈ M is not a cut point of p if there exist positivenumber δ > 0, a point q ′ ∈ M and a minimizing geodesic γ : [ 0, d(p, q)+ δ ] −→ M thatsatisfies γ (0) = p, γ (d(p, q)) = q , and γ (d(p, q)+ δ) = q ′. We define
Cut(p) := {q ∈ M |q is a cut point of p} ∪ {p}and call it the cut locus of p.
Remark 3 According to [11, Proposition 5.10], the distance function rp(·) := d(p, · ) for afixed point p ∈ M is C1 on Mp := M − Cut(p) and the Hausdorff dimension of Cut(p)is smaller than n since we have assumed that (M,D, d) does not admit nontrivial singularminimizers. According to [24, Theorem 2.17], if the subbundle D is regular at any pointp ∈ M , then the Hausdorff dimension of a sub-Riemannian manifold (M,D, d) is greaterthan or equal to n and the Lebesgue measure μ on M is absolutely continuous with respectto the Hausdorff measure, so that μ(Cut(p)) = 0.
Finally we mention the measure contraction property.
Definition 6 Let μ be a Lebesgue measure on the manifold M . For any point p ∈ M ,we define the contraction map Φp( ·, · ) : [0, 1] × Mp −→ M byΦp(t, q) := expq
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( (t − 1) d(r2p)q ), where d(r2
p)q is the exterior derivative of the function r2p at q . For a
real number κ ∈ R, we define a function sκ (·) by
sκ (r) :=⎧⎨⎩
sin(√κr)/
√κ if κ > 0,
r if κ = 0,sinh(
√|κ|r)/√|κ| if κ < 0.
We fix two arbitrary real numbers N ∈ [1,+∞) and κ ∈ R. We say that the triple (M, d, μ)satisfies MCP(κ, N ) if it satisfies that for any point p ∈ M , any real number t ∈ (0, 1] andany nonnegative measurable function f : M −→ [0,+∞), we have
∫
Φp(t,Mp)
f ◦Φp(t, ·)−1(q)dμ(q) ≥∫
M
t sκ (tr p(q))N−1
sκ (rp(q))N−1 f (q)dμ(q),
where f has compact support contained in M − {p} and if κ > 0, then its support is alsocontained in the open metric ball B(p, π/
√κ) centered at p of radius π/
√κ .
Remark 4 (1) Φp( ·, q) : [0, 1] −→ M is the constant speed unique minimizing geodesicfrom p to q .
(2) For any real number t ∈ (0, 1], Φp(t, ·) : Mp −→ M is injective since there are nonontrivial singular geodesics (see the proof of Theorem 1 in [18]).
(3) If a triple (M, d, μ) is an n-dimensional Riemannian manifold M with the Riemanniandistance function d and the Riemannian measure μ, then MCP(κ, n) is equivalent to thecondition Ric ≥ κ , where Ric is the Ricci curvature of M [26].
3 Laplacian comparison theorem
In this section, we prove the Laplacian comparison theorem that we need to prove Theorems2 and 3.
Definition 7 For a C1-function u on (M,D, d), we define a vector field ∇u on M that istangent to D by 〈∇u, X〉 := Xu for any vector field X on M that is tangent to D.
We set cotκ (·) := s′κ (·)/sκ (·) for the function sκ (·) defined in Definition 6.
Proposition 1 (Laplacian comparison theorem) Let a triple (M,D, d) be a sub-Riemannianmanifold andμ a Lebesgue measure on M. We assume that (M, d, μ) satisfies MCP(κ, N ) forsome real numbers κ ∈ R and N ∈ [1,∞). Then for any point p ∈ M and any nonnegativeLipschitz function f whose support is compact and contained in M − {p}, we have
∫
M
〈∇rp,∇ f 〉dμ ≥∫
M
{−(N − 1) cotκ ◦rp} f dμ. (1)
Proof The proof follows the same idea of the proof of Proposition 3.1 in [22]. Let p beany point in M and f be any nonnegative Lipschitz function whose support is compact andcontained in M − {p}. Since f is a Lipschitz function, there exists ∇ f (q) for μ-a.e. q ∈ M[25, Theorem 2.5]. From
⋃0<t<1Φp(t,Mp) = Mp and t r p(Φp(t, · )−1(q)) = rp(q) for
any number t ∈ (0, 1] and any point q ∈ Mp , we have
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⟨∇rp(q),
d
dtΦp(t, ·)−1(q)
∣∣∣t=1
⟩
= d
dtrp(Φp(t, ·)−1(q))
∣∣∣t=1
= −rp(q).
Note also that | ∇rp(q) | = 1 and | ddt Φp(t, · )−1(q) |t=1 | = rp(q) for any point q ∈ Mp .
Then we have
d
dtΦp(t, ·)−1(q)
∣∣∣t=1
= −rp(q)∇rp(q).
By these calculations, we have∫
M
〈rp∇rp,∇ f 〉 dμ
= −∫
Mp
⟨d
dtΦp(t, · )−1(q)
∣∣∣t=1, ∇ f (q)
⟩dμ(q)
= −∫
Mp
d
dtf ◦Φp(t, · )−1(q)
∣∣∣t=1
dμ(q)
= limt→1−0
⎡⎢⎣
∫
Mp
f ◦Φp(t, · )−1(q)
1 − tdμ(q)−
∫
Mp
f (q)
1 − tdμ(q)
⎤⎥⎦ .
The last equality follows from the Lebesgue dominated convergence theorem. We give anonnegative integrable function that bounds the sequence of functions from above as in thefollowing.
We define a function Ft : M −→ R by
Ft (q) :=⎧⎨⎩
f ◦Φp(t, ·)−1(q)− f (q)
1 − tif q ∈ Mp ,
0 if q ∈ M − Mp .
We take L as a Lipschitz constant of f . We have |Ft (q)| ≤ Lrp(q)/t ≤ 2Lrp(q) fort sufficiently near 1 since d(q, Φp(t, · )−1(q)) = (1 − t) rp(q)/t . The function rp(·) isintegrable on the support of f .
Since we assume that the triple (M, d, μ) satisfies MCP(κ, N ), we have
limt→1−0
⎡⎢⎣
∫
Mp
f ◦Φp(t, · )−1(q)
1 − tdμ(q)−
∫
Mp
f (q)
1 − tdμ(q)
⎤⎥⎦
≥ limt→1−0
⎡⎣
∫
M
t sκ (t r p(q)) N−1 f (q)
(1 − t) sκ (rp(q)) N−1 dμ(q) −∫
M
f (q)
1 − tdμ(q)
⎤⎦
=∫
M
− d
dt{t sκ (t r p(q))
N−1}∣∣∣t=1
f (q)
sκ (rp(q)) N−1 dμ(q)
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=∫
M
{−1 − (N − 1) rp(q) cotκ (rp(q))} f (q) dμ(q).
Note that 〈rp∇rp,∇ f 〉 = 〈∇rp,∇(rp f )〉 − f for μ-almost everywhere since ∇(rp f ) =f ∇rp + rp∇ f . We have∫
M
〈∇rp,∇(rp f )〉 dμ ≥∫
M
{−(N − 1) cotκ (rp(q))} rp(q) f (q) dμ(q). (2)
We now define a function f : M −→ R for any nonnegative Lipschitz function f : M −→ R
whose support is compact and contained in M − {p} as
f (q) :=⎧⎨⎩
f (q)
rp(q)if q = p,
0 if q = p.
Then f is also a Lipschitz continuous function whose support is compact and contained inM − {p}. If we take this f as f in the Eq. (2), we obtain the Laplacian comparison theorem(1). ��
4 Maximum principle
In this section, we provide some analytic tools and the maximum principle needed in the proofof Theorem 2. Let U be any open subset of the sub-Riemannian manifold (M,D, d) and μa Lebesgue measure on M . In the following, (M, d, μ) means a triple of a sub-Riemannianmanifold and a Lebesgue measure. We denote by L2(U ), the set of all square integrablefunctions on U , and by C∞
0 (U ), the set of all smooth functions with compact supports onU . If a function u ∈ L2(U ) satisfies
∫U 〈∇u,∇u〉 dμ < +∞, we call u a W 1,2-function,
where the gradient is associated with the sub-Riemannian metric (see Definition 7) and thederivative is in the following distributional sense [13]. For any vector field X on U tangentto D and any function u ∈ L2(U ), we define a measurable function Xu by∫
U
(Xu) φ dμ = −∫
U
u (X∗ φ) dμ
for any φ ∈ C∞0 (U ), where X∗ is the formal adjoint operator of X . We denote by W 1,2(U )
the set of all W 1,2-functions on U .
Definition 8 (Locally W 1,2-function) A measurable function u : M −→ R is a locallyW 1,2-function if for any point p ∈ M , there exists an open neighborhood U of p such thatu|U ∈ W 1,2(U ).
Definition 9 (μ-subharmonic function) A locally W 1,2-function u : M −→ R is a μ-subharmonic function if
∫M 〈∇u,∇ f 〉 dμ ≤ 0 for any nonnegative smooth function f :
M −→ R with compact support.
The purpose of this section is to prove the following.
Proposition 2 (Maximum principle) Let a triple (M,D, d) be a complete sub-Riemannianmanifold. We assume that the triple (M, d, μ) satisfies MCP(κ, N ) for real numbers κ ∈R, N ∈ [1,+∞). If a continuousμ-subharmonic function u : M −→ R attains its maximumvalue, then u is constant on M.
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We use Kuwae’s maximum principle [21, Theorem 8.5] to prove Proposition 2. Firstwe recall some definitions [12]. Let E be a nonnegative definite symmetric bilinear formdefined on a subspace D[E] ⊂ L2(M, μ). D[E] is a dense subspace of L2(M, μ). Wecall E simply a symmetric form. Let (·, ·)L2 be the inner product on L2(M, μ). We defineE1(u, v) := E(u, v)+ (u, v)L2 for any u, v ∈ D[E]. E1 is a metric on D[E]. We say that E isclosed if D[E] is complete with respect to E1.
Definition 10 We say that a symmetric form E is closable if the following holds. If a sequence{un}n=1,2,... ⊂ D[E] satisfies E(un −um, un −um) → 0 as n,m → +∞ and (un, un)L2 → 0as n → +∞, then E(un, un) → 0 as n → +∞.
Let A be the set of all E1-Cauchy sequences. Two sequences {un}, {vn} ∈ A are regardedto be equivalent if E1(un − vn, un − vn) → 0 as n → ∞. We denote by F , the set of allequivalence classes. If a symmetric form E is closable, we have a closed symmetric form byextending E on F .
Definition 11 We say that a symmetric form E is Markovian if for each ε > 0, there existsa real function φε : R −→ R satisfying the following two conditions.
(1) φε(t) = t for any t ∈ [0, 1], −ε ≤ φε(t) ≤ 1 + ε for any t ∈ R and0 ≤ φε(s)− φε(t) ≤ s − t whenever t < s.
(2) For any u ∈ D[E], we have φε(u) ∈ D[E] and E(φε(u), φε(u)) ≤ E(u, u).
We say that a symmetric form (E,D[E]) is a Dirichlet form if it is closed and Markovian.We denote by C0(M), the set of all continuous functions on M with compact support.
Definition 12 We say that a symmetric form E is regular if it posseses a core, where a coreof E is a subset of D[E] ∩ C0(M) such that it is dense in D[E] with respect to the E1-normand dense in C0(M) with respect to the uniform norm.
Definition 13 We say that a symmetric form E is strongly local if E(u, v) = 0 for anyu, v ∈ D[E] such that both of the supports of u and v are compact and v is constant on aneighbourhood of the support of u.
Let (E,D[E]) be a regular strongly local Dirichlet form. According to [12], there existsa diffusion process on M associated to (E,D[E]). We denote by {Tt }t>0, the associatedsemigroup on L2(M, μ). We define some classes of functions.
C(M) := { f | f is a continuous function on M},C∞(M) := { f | f is a continuous function on M vanishing at infinity},Bb(M) := { f | f is a bounded Borel measurrable function on M},Cb(M) := { f | f is a bounded continuous function on M}.
Definition 14 We say that {Tt }t>0 has Feller property if Tt (C∞(M)) ⊂ C∞(M) for anyt > 0 and if ||Tt u − u|| → 0 as t → 0 for any u ∈ C∞(M), where || · || is uniform norm onC∞(M).
Definition 15 We say that {Tt }t>0 has strong Feller property if Tt (Bb(M)) ⊂ Cb(M) forany t > 0.
{Tt }t>0 is said to have doubly Feller property if it has both Feller property and strongFeller property.
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Definition 16 (see Definition 3.2 of [21]) A measurable function u : X −→ R is locally anelement of F if there exist a sequence {un}n=1,2,... ⊂ F and an open covering {En}n=1,2,...
of M such that En ⊂ En+1 for n = 1, 2, . . . and u = un μ-a.e. on En for n = 1, 2, . . . .
Definition 17 An E-subharmonic function u : X −→ R is a measurable function that islocally an element of F and satisfies E(u, v) ≤ 0 for any v ∈ F such that v is a nonnegativeand continuous function whose support is compact [21].
Kuwae’s maximum principle is stated as follows.
Theorem 5 (see Theorem 8.5 of [21]) Let (X, d) be a locally compact separable metricspace and m a positive Radon measure with full topological support. Consider a regularstrongly local symmetric Dirichlet form (E,F) on L2(X,m). Let G be an open connectedsubset of X. We assume that (E,F) is associated with a doubly Feller m-symmetric diffusionprocess admitting a jointly continuous heat kernel pt (x, y) with respect to m. If a continuousE-subharmonic function u : G −→ R attains its maximum at a point x0 ∈ G, then we haveu ≡ u(x0).
We provide the Dirichlet form associated with the sub-Riemannian structure. For a triple(M, d, μ), we define a symmetric bilinear form (E,D[E]) on the Hilbert space L2(M, μ) byD[E] := C∞
0 (M) and E(u, v) := ∫M 〈∇u,∇v〉 dμ for any u, v ∈ C∞
0 (M). Note that C∞0 (M)
is dense in L2(M, μ) and that E is regular, strongly local, closable, and Markovian. We extendthe domain D[E] to F by the same way as above. Then (E,F) is a regular strongly localsymmetric Dirichlet form. If (M,D, d) is a complete metric space, the semigroup associatedwith (E,F) has a heat kernel pt (x, y) that is continuous with respect to x, y ∈ M ([31],[32, §12], [33, Proposition 3.1]). We obtain that a μ-subharmonic function in Definition 9 isE-subharmonic from the following lemma.
Lemma 1 A locally W 1,2-function (see Definition 8) is locally an element of F .
Proof First we remark that W 1,2(M) = F if (M,D, d) is complete. The proof of this fact isin the same way as in the Riemannian case [16, Theorem 2.7]. A sub-Riemannian analogueof the Rademacher theorem is given in [25, Theorem 2.5] and the Meyers-Serrin theorem isgiven in [13]. Let u be a locally W 1,2-function and p ∈ M a point. We choose a sequenceof open balls {B(p, l)}l=1,2,... that cover M . The closure of each ball is compact from thecompleteness of (M,D, d). This fact follows from [2, Theorem 2.11]. For each l, there existsa relatively compact open neighborhood Ul around p such that B(p, l) ⊂ Ul and u|Ul is aW 1,2-function. Then we cut off u and obtain a function ul ∈ W 1,2(M) such that ul = u onB(p, l) and ul = 0 on M − Ul for each l. Hence u is locally an element of F in the sense ofDefinition 16 since W 1,2(M) = F . ��
In the following, we use some results of analysis on a metric measure space to check thatif (M, d, μ) satisfies MCP(κ, N ), then the semigroup {Tt }t>0 of the Dirichlet form (E,F)associated with (M, d, μ) has doubly Feller property.
Definition 18 (see [28]) For a triple (M, d, μ) and the contraction map Φp defined in Def-inition 6, (M, d, μ) satisfies the strong doubling condition if there exists a constant b > 0such that for μ-almost every point p ∈ M , any Borel measurable subset A ⊂ M , and anyreal number t ∈ [1/2, 1], we have
μ(Φp(t, ·)−1(A) ) ≤ bμ(A).
Lemma 2 The strong doubling condition follows from MCP(κ, N ) for κ ≥ 0 and locallyfrom MCP(κ, N ) for κ < 0.
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Proof In the case of κ = 0, for any point p ∈ M , any real number t ∈ (0, 1] and any non-negative measurable function f : M −→ [0,+∞) whose support is compact and containedin M − {p}, we have
∫
M
t N f (q) dμ(q) ≤∫
Φp(t,Mp)
f ◦Φp(t, · )−1(q) dμ(q)
from Definition 6. This is equivalent to that for any Borel measurable subset A ⊂ M , anypoint p ∈ M , and any real number t ∈ (0, 1], we have
t Nμ( A) ≤ μ(Φp(t, A)).
If we take A = Φp(t, · )−1(A) and b = 2N , then this inequality is the strong doublingcondition in Definition 18. In the case of κ > 0, MCP(κ, N ) implies MCP(0, N ) so that weobtain the strong doubling condition. In the case of κ < 0, from Definition 6, as in the caseof κ = 0, we have
∫
Φp(t, · )−1(A)
t sκ (t r p(q))N−1
sκ (rp(q))N−1 dμ(q) ≤ μ(A),
for any Borel measurable subset A ⊂ M , any point p ∈ M , and any real number t ∈(0, 1]. We consider any open metric ball B and any Borel subset A such that A ⊂ B andΦp(t, · )−1(A) ⊂ B. Since the closed ball B is compact and the function
t sκ (t r p(q))N−1
sκ (rp(q))N−1
is continuous with respect to t ∈ [1/2, 1] and q ∈ B, there exists a positive number b > 0such that
1
b≤ t sκ (t r p(q))N−1
sκ (rp(q))N−1
for any t ∈ [1/2, 1] and q ∈ B. Therefore we have
μ(Φp(t, · )−1(A)) ≤ bμ(A)
for any t ∈ [1/2, 1], namely the strong doubling condition on B. ��Definition 19 A measurable function g : M −→ R is an upper gradient of a functionu : M −→ R if for any absolutely continuous curve γ : [0, 1] −→ M satisfying l(γ |[s,t]) =|t − s| l(γ ) for any 0 ≤ s < t ≤ 1, where l means the length of the curve, we have
|u(γ (1))− u(γ (0))| ≤1∫
0
g ◦ γ (t) dt.
Theorem 6 (see [28]) We assume that (M, d, μ) satisfies the strong doubling condition andg is an upper gradient of a measurable function u : M −→ R. Then there exists a constantCP > 0 independent of u such that for any point a ∈ M and any real number r > 0, wehave
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188 Geom Dedicata (2014) 168:177–196
−∫
B(a,r)
∣∣∣∣∣∣∣u − −
∫
B(a,r)
u dμ
∣∣∣∣∣∣∣dμ ≤ CP r −
∫
B(a,3r)
g dμ,
where barred integral signs mean integral averages.
We assume that (M, d, μ) satisfies MCP(κ, N ) for some real numbers κ ∈ R and N ∈[1,+∞). For any function u ∈ C1(M), |∇u| is an upper gradient of u, so that we have
−∫
B(a,r)
∣∣∣∣∣∣∣u − −
∫
B(a,r)
u dμ
∣∣∣∣∣∣∣dμ ≤ CP r −
∫
B(a,3r)
|∇u| dμ
from Lemma 2 and Theorem 6. We extend this inequality for any element u of F by thestandard discussion. Using the Schwartz inequality, we have a weak (1, 2)-Poincaré inequality
−∫
B(a,r)
∣∣∣∣∣∣∣u − −
∫
B(a,r)
u dμ
∣∣∣∣∣∣∣dμ ≤ CP r
⎛⎜⎝ −
∫
B(a,3r)
|∇u|2 dμ
⎞⎟⎠
12
.
Note also that the following doubling condition follows from the strong doubling condition.There exists a real number N ∈ [1,+∞) such that for any real number r > 0 and any pointa ∈ M , we have
μ(B(a, 2r)) ≤ 2Nμ(B(a, r)).
From [15, Theorem 5.1, Corollary 9.8], we have the following theorem.
Theorem 7 If (M, d, μ) satisfies MCP(κ, N ) for some real numbers κ ∈ R and N ∈[1,+∞), then there exists a positive number R > 0 such that a weak (2, 2)-Poincaré inequal-ity holds
⎛⎜⎝ −
∫
B(a,r)
∣∣∣∣∣∣∣u − −
∫
B(a,r)
u dμ
∣∣∣∣∣∣∣
2
dμ
⎞⎟⎠
12
≤ CP r
⎛⎜⎝ −
∫
B(a,r)
|∇u|2 dμ
⎞⎟⎠
12
for any u ∈ F and any positive number r > 0 smaller than R. If κ ≥ 0, then R = +∞.
Proof First we prove this theorem assuming that N > 2 for applying Theorem 5.1 andCorollary 9.8 in [15]. The other case follows from the fact that if N1 < N2 then MCP(κ, N2)
follows from MCP(κ, N1). ��For the proof of Proposition 2, we recall the definition of strong regularity [33]. Let (E,F)
be the Dirichlet form associated with the sub-Riemannian structure. There exists a positivesemidefinite, symmetric bilinear form Γ on F with values in the signed Radon measures onM , defined by the formula [33],∫
M
φ dΓ (u, u) = E(u, φ u)− 1
2E(u2, φ),
for every u ∈ F ∩ L∞(M, μ) and every φ ∈ F ∩ C0(M). We define a class of functions,
Floc := {u ∈ L2
loc(M, μ) | Γ (u, u) is a Radon measure},
and a pseudo metric on M ,
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Geom Dedicata (2014) 168:177–196 189
ρ(x, y) := sup { u(x)− u(y) | u ∈ Floc ∩ C(M), Γ (u, u) ≤ μ as measures on M}.ρ is called intrinsic metric.
Definition 20 (E,F) is said to be strongly regular if intrinsic metric ρ is a metric on Mwhose topology coincides with the original one.
Proof (Proof of Proposition 2) According to [33, Theorem 3.5], if (E,F) is strongly regular,the doubling condition and a weak (2, 2)-Poincaré inequality implies the parabolic Harnackinequality. Moreover, the parabolic Harnack inequality implies the doubly Feller property[21, Theorem 8.5]. We check the strong regularity later and continue the proof of proposition2. Let u : M −→ R be a continuousμ-subharmonic function that attains its maximum value.u is E-subharmonic (see Definition 17), so that it is a constant function from Theorem 5 inthe case of κ ≥ 0. In the case of κ < 0, we use Theorem 5 only on an open metric ball, sothat u is constant on some ball. Since M is connected, u is globally constant.
Finally, we check the strong regularity. It is a well-known fact that the sub-Riemanniandistance between two points p and q defined in Sect. 2 coincides with sup{u(p)−u(q) | u ∈C∞
0 (M), |∇u| ≤ 1} (see § 2.3 of [2]). Though in general, this is smaller than intrinsic metricρ, the two distance functions coincide in our setting. The proof of this is in the same way asin the Riemannian case, i.e., any continuous function in W 1,2(M) is approximated at p andq by a sequence of smooth functions {ui }i=1,2,... satisfying that |∇ui | ≤ 1 and their supportsare compact [13,16]. Therefore, our Dirichlet form (E,F) satisfies the strong regularity in[33]. ��
5 Proof of the main theorems
We say that a curve γ : R −→ M is a straight line if for any two real numbers a < b, wehave d(γ (a), γ (b)) = |b − a|. If the domain of γ is [0,+∞), we call γ a ray. For a ray (or astraight line), we define the Busemann function bγ by bγ (p) := limt→+∞( t − d(p, γ (t)) )for any point p ∈ M . First we prove the smoothness of the Busemann function. We takea sequence of nonnegative numbers {ti }i=1,2,... ⊂ [0,+∞) satisfying limi→+∞ ti = +∞.Let γ be a ray parametrized by the arc length. We note that the Busemann function bγ ofγ is a Lipschitz function. By an analogue of the Rademacher theorem on a sub-Riemannianmanifold [25], there exist ∇rγ (ti )(p) and ∇bγ (p) for all i = 1, 2, . . . and μ-almost everypoint p ∈ M . We fix such a point p. p and γ (ti ) are connected by a minimizing geodesicσi : [0, d(p, γ (ti ))] −→ M for each i = 1, 2, . . . , where |σi | ≡ 1, σi (0) = p andσi (d(p, γ (ti ))) = γ (ti ). We take a real number T > 0. From the completeness of M and theAscoli-Arzelà theorem [4, Theorem 2.5.14], there are a continuous curve σ : [0, T ] −→ Mand a subsequence {σ j } j=1,2,... of {σi }i=1,2,... such that σ j uniformly converges to σ asj → +∞.
We define the length of a continuous curve c : [0, T ] −→ M of a sub-Riemannianmanifold (M,D, d) by l(c) := sup
∑Ni=1 d(c(ti−1), c(ti )), where the supremum is given by
all partitions 0 = t0 < t1 < · · · < tN = T of [0, T ]. According to [3, § 2.3], l gives a lowersemicontinuous functional on the space of all continuous curves C([0, T ],M). We note thatif c is absolutely continuous and satisfies c(t) ∈ Dc(t) for almost every t ∈ [0, T ], then we
have l(c) = ∫ T0 |c(t)| dt . From these facts, we have
l(σ ) ≤ lim infj→+∞ l(σ j ) = T .
Since d(p, σ (T )) = lim j→+∞ d(p, σ j (T )) = T, σ is a minimizing geodesic.
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190 Geom Dedicata (2014) 168:177–196
Lemma 3 We have
bγ ◦ σ(t) = t + bγ ◦ σ(0)for any t ∈ [0, T ].
The proof of this lemma is in the same way as in [30, Theorem 3.8.2(3)]. Since we assumethat the sub-Riemannian manifold (M,D, d) does not contain nontrivial singular minimizers,the following holds.
Lemma 4 (see Lemma 1 in [18]) For any point p ∈ M, a local coordinate U around p anda compact subset K ⊂ U − {p}, we denote by K the set of all α ∈ T ∗
p M that satisfy that
expp (α) ∈ K and expp(·α) : [0, 1] −→ M is a minimizing geodesic. Then K is compact.
From this lemma, we have the following.
Lemma 5 If (M, d, μ) satisfies MCP(0, N ) for some N ∈ [1,+∞), then the Busemannfunction bγ of any ray γ : [0,+∞) −→ M is μ-subharmonic.
Proof We take sufficiently small real numbers 0 < R1 < R2 and an annulus A := {q ∈M | R1 ≤ d(p, q) ≤ R2} as K in Lemma 4. Let δ be a real number such as R1 ≤ δ ≤ R2.Let ξ j (t) be a solution of the Hamiltonian equation on T ∗M such that π ◦ ξ j (t) = σ j (t) forany t ∈ [0, d(p, γ (t j ))], where σ j is a minimizing geodesic parametrized by the arc lengthconnecting p and γ (t j ). Then expp δξ j (0) for each j = 1, 2, . . . is contained in the annulusA since (ξ j (0), ξ j (0))p = 1. From Lemma 4, the sequence {δξ j (0)} j=1,2,... ⊂ T ∗
p M has aconverging sequence {δξk(0)}k=1,2,... ⊂ T ∗
p M . Let δξ0 ∈ T ∗p M be the limit of {δξk(0)}k=1,2,...
and ξ(t) an integral curve of H with ξ0 as the initial value, where H is the Hamiltonian vectorfield on T ∗M . We have ξk(t) → ξ(t) as k → +∞ for every t ∈ [0, T ], since a solution ofξ (t) = Hξ(t) depends continuously on the initial value. Since π ◦ ξk(t) → σ(t) as k → +∞and the projection map π is continuous, we have
π ◦ ξ(t) = σ(t).
This implies that σk(0) → σ (0) as k → +∞. Note that bγ ◦ σ(t) = t + bγ ◦ σ(0) forany t ∈ [0, T ] by Lemma 3 and bγ is a 1-Lipschitz function. Then we have |∇bγ (p)| =1, |σ (0)| = 1 and 〈∇bγ (p), σ (0)〉 = d
dt bγ ◦ σ(t)|t=0 = 1 so that ∇bγ (p) = σ (0). Wehave also −∇rγ (tk )(p) = σk(0) since d
dt rγ (tk ) ◦σk(t)|t=0 = −1 and |σk(0)| = 1. From thesefacts, we have
−∇rγ (tk )(p) → ∇bγ (p)
as k → +∞. By Proposition 1 for κ = 0, for any nonnegative Lipschitz function f : M −→[0,+∞) with compact support, we have∫
M
〈∇bγ ,∇ f 〉 dμ = − limk→+∞
∫
M
〈∇rγ (tk ),∇ f 〉 dμ
≤ (N − 1) limk→+∞
∫
M
f
rγ (tk )dμ
= 0.
bγ is a locally W 1,2-function since |∇bγ (p)| = 1 forμ-almost every p ∈ M . This completesthe proof. ��
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Geom Dedicata (2014) 168:177–196 191
Remark 5 Though Lemma 5 is an analogue of [22, Lemma 4.6], we cannot prove it inthe same way as in [22, Lemma 4.6]. The essential difference is how to prove that thesequence {−∇rγ (tk )(p)}k=1,2,... approximates ∇bγ (p). In [22, Lemma 4.6], the convergenceof the sequence of the initial vectors of the minimizing geodesics follows directly from theconvergence of the sequence of the minimizing geodesics, since the set of the vectors whoselength are 1 in Tp M is compact. In the sub-Riemannian case, we need Lemma 4 because themetric (·, ·)p on T ∗
p M is degenerate in general and the set of the covectors whose length are1 in T ∗
p M is not compact. Lemma 4 does not hold if (M,D, d) contains a nontrivial singularminimizer.
Theorem 8 (see [17,19]) If a set of vector fields X1, . . . , Xm on a domainΩ of Rn satisfies
Hörmander’s condition (Definition 1), then the operator −∑mi=1 X∗
i Xi is hypoelliptic onΩ .
We use this theorem to prove the following lemma.
Lemma 6 If (M, d, μ) is complete and satisfies MCP(0, N ) for some N ∈ [1,+∞), thenthe Busemann function bγ of any straight line γ is smooth.
for any s ∈ R. By Lemma 5, b+ + b− is μ-subharmonic, hence from Proposition 2, we haveb+ + b− ≡ 0 on M . Since b− is also μ-subharmonic from Lemma 5, we have∫
M
〈 ∇bγ ,∇ f 〉 dμ = 0
for any nonnegative smooth function f : M −→ [0,+∞) with compact support, namelyb+ and b− are μ-harmonic. For any point p ∈ M , we choose a sufficiently small localcoordinate (U, φ) around p. φ : U −→ R
n is the coordinate map. Let X1, . . . , Xm be aset of vector fields on φ(U ) that is a orthonormal frame of the subbundle D on φ(U ) withrespect to the metric 〈·, ·〉 of D. Then the function bγ ◦ φ−1 on φ(U ) is a harmonic function,i.e., −∑m
i=1 X∗i Xi (bγ ◦ φ−1) = 0 with respect to the Lebesgue measure on φ(U ) in the
distributional sense. By Theorem 8, bγ is smooth. ��Finally we prove the main theorems.
Proof (Proof of Theorem 2)We have |∇bγ | ≡ 1 on M since bγ is smooth and |∇bγ (p)| = 1for almost every p ∈ M . By the completeness of (M,D, d), ∇bγ gives a 1-parameter group
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192 Geom Dedicata (2014) 168:177–196
of transformations on M and a diffeomorphism between b−1γ (0)× R and M . We prove that
any integral curve of ∇bγ is a straight line in the same way as in the Riemannian case [29,Lemma 3.10]. ��Proof (Proof of Theorem 3) We assume that there exist a metric space X and an isometryφ : (X × R, dX×R) −→ (M,D, d), where dX×R denotes the product distance function. Wefix any point x0 ∈ X and define two straight lines σ : R −→ X × R and γ : R −→ Mby σ(t) := (x0, t) and γ (t) := φ(x0, t). By Theorem 2, the gradient vector field ∇bγ forγ gives a 1-parameter group on M and a diffeomorphism between b−1
γ (0) × R and M . X
endows with the same differential structure as each b−1γ (t) by φ−1(b−1
γ (t)) = X × {t} forany t ∈ R, since for any (y, s) ∈ X × R, we have bσ (y, s) = s and
bγ ◦ φ(y, s) = limt→+∞(t − d(φ(y, s), γ (t)))
= limt→+∞(t − d(φ(y, s), φ ◦ σ(t)))
= limt→+∞(t − dX×R((y, s), σ (t)))
= bσ (y, s).
Hence, (X × R, dX×R) is given a product manifold structure by φ and a 1-parametergroup associated with ∇bγ so that it is identified with the sub-Riemannian manifold(M,D, d).
Let t0 ∈ R be a real number. We define ψt : X × {t0} −→ X × {t} by ψt (x, t0) := (x, t)for any (x, t) ∈ X × R. We note that the subbundle D splits as D(x,t) = (T X ∩ D)(x,t) ⊕ R
for any (x, t) ∈ X × R since any straight line in M is tangent to D. ��Lemma 7 The linear map (ψt )∗|( T X∩ D)(x,t0) gives an isomorphism between (T X ∩D)(x,t0)and (T X ∩ D)(x,t) for any (x, t) ∈ X × R.
We prove this lemma later and continue the proof of Theorem 3. Let S be a sectionof (T X ∩ D)|X×{t0}. We extend S to the vector field S on X × R tangent to T X ∩ D byS(x,t) := (ψt )∗Sx for any (x, t) ∈ X × R. Then we have [ ∂
∂t , S ](x,t) = 0 ∈ D(x,t) for any(x, t) ∈ X × R since (ψt )∗S does not move along the integral curves of ∂
∂t . By the condition(∗∗) for p = (x1, t1) ∈ X × R and v = ( ∂
∂t )(x1,t1), there exists a vector field W on X × R
tangent to T X ∩ D such that [ ∂∂t , W ](x1,t1) /∈ D(x1,t1). From Lemma 7, there exists a sec-
tion S of (T X ∩ D)|X×{t0} such that W(x1,t1) = S(x1,t1). This implies that [ ∂∂t , W ](x1,t1) ∈
D(x1,t1) since [ ∂∂t , S ](x1,t1) = 0 ∈ D(x1,t1) and by the tensorial property of the curva-
ture in the sense of [24, §4.1]. This is a contradiction. There does not exist such a metricspace X .
Proof (Proof of Lemma 7) It is sufficient to prove that
(ψt )∗(v) ∈ D(x,t)
for any (x, t) ∈ X × R and any v ∈ (T X ∩ D)(x,t0), since (ψt )∗ is nondegenerate and dim(T X ∩ D)(x,t0) = dim (T X ∩ D)(x,t). Let δ > 0 be a positive number and c : [0, δ) −→X × {t0} a smooth curve satisfying c(0) = (x, t0) ∈ X × {t0}, c(0) = v and c(s) ∈ Dc(s) forany s ∈ [0, δ). We take a positive number T ∈ (0, δ) and a constant C := max s∈[0,T ] |c(s)|,then we have
d((x, t0), c(s)) ≤s∫
0
|c(τ )| dτ ≤ Cs
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Geom Dedicata (2014) 168:177–196 193
for any s ∈ (0, T ]. We suppose that (ψt )∗(v) /∈ D(x,t). Then since dds |s=0 ψt ◦ c(s) /∈ D(x,t)
and by [10, Lemma 7], there exists a sequence of positive numbers {sn}n=1,2,... such thatlimn→+∞ sn = 0 and
d((x, t), ψt ◦ c(sn)) ≥ nsn
for n = 1, 2, . . . . Therefore, we have
C ≥ d((x, t0), c(sn))
sn= d((x, t), ψt ◦ c(sn))
sn≥ n
for any n = 1, 2, . . . since ψt : X × {t0} −→ X × {t} is an isometry. This is a contradiction.We obtain that (ψt )∗(v) ∈ D(x,t). ��Remark 6 The condition (∗∗) is essential for Theorem 3. If we do not assume the condition(∗∗), then there exists an example of sub-Riemannian manifolds that split isometrically andsatisfy the condition (∗) except the non-existence condition of nontrivial singular minimizers.Let (M1,D1, d1) and (M2,D2, d2) be two sub-Riemannian manifolds and consider a pair(M1×M2,D1⊕D2), where D1⊕D2 is the direct sum of D1 and D2. This satisfies Hörmander’scondition and the sub-Riemannian distance function determined by D1 ⊕ D2 coincides withthe product distance function of d1 and d2. D1 ⊕ D2 does not satisfy the condition (∗∗).According to [27], for two metric measure spaces (X1,m1) and (X2,m2) satisfying themeasure contraction property, (X1 × X2,m1 × m2) also satisfies the measure contractionproperty, where m1 × m2 means the product measure. R satisfies MCP(0, 1) with respectto the Lebesgue measure and any Heisenberg group H
n satisfies the measure contractionproperty with respect to the Lebesgue measure [20]. Hence the sub-Riemannian manifoldH
n × R satisfies the measure contraction property with respect to the Lebesgue measure andsplits isometrically. However H
n × R contains nontrivial singular minimizers. For any pointp ∈ H
n , any segment of the straight line c(t) := (p, t) ∈ Hn × R is a singular curve [24,
Theorem 5.3]. Nevertheless, Theorem 2 holds for Hn × R. In the proof of Theorem 2 for M ,
we need the non-existence condition of nontrivial singular minimizers for the following (1),(2) and (3).
(1) μ(Cut(p)) = 0 for any point p ∈ M (Remark 3).(2) The contraction map Φp(t, ·) : Mp −→ M for any point p ∈ M in Remark 4 (2) is
injective.(3) Lemma 4.
For M = Hn ×R, (1) and (2) follow from the fact that a curve γ : [0, T ] −→ H
n ×R, γ (t) =(γ1(t), γ2(t)) ∈ H
n ×R is a minimizer if and only if both of γ1 and γ2 are minimizers. Lemma4 is used in the proof of Lemma 5. Lemma 5 for M = H
n ×R is proved by using [18, Lemma1] instead of Lemma 4. We note that every nontrivial minimizer in H
n × R except segmentsof the straight lines of R-direction is not a singular curve [24, Lemma 5.5]. Hence Theorem2 holds for H
n × R.If (1), (2) and Lemma 5 hold for a triple (M, d, μ), then we obtain Theorems 2 and 3 for
(M, d, μ) even if nontrivial singular minimizers exist.
Proof (Proof of Corollary 1) Let M be the universal covering space of M . M is endowedwith the sub-Riemannian structure D induced from D. Then (M, D, d) has nonnegative gen-eralized Ricci curvature in the sense of [1] and hence satisfies MCP(0, 5) for the appropriatemeasure appeared in Lemma 4.2 of [1]. (M, D, d) contains a straight line since M is com-pact and π1(M) is an infinite group. From Theorem 2, there exists a 2-dimensional smoothmanifold N such that M is diffeomorphic to N × R. Since N is simply connected, it isdiffeomorphic to either S2 or R
2. ��
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194 Geom Dedicata (2014) 168:177–196
6 Example
In this section we treat an example of sub-Riemannian manifolds satisfying the assumptionsof Theorems 2 and 3.
Let M = R3, X1 = (1, 0,−y/2), X2 = (0, 1, x/2) and D be the subbundle generated by
X1 and X2. D satisfies Hörmander’s condition. D endows with the metric so that {X1, X2}gives an orthonormal frame. We denote the sub-Riemannian distance function associatedwith D by d . We call the triple (R3,D, d) the 3-dimensional Heisenberg group and denote itby H. The operation on H as a group is defined by
(x1, y1, z1) · (x2, y2, z2) :=(
x1 + x2, y1 + y2, z1 + z2 + 1
2(x1 y2 − x2 y1)
).
H is a Lie group and d is the left invariant distance function with respect to this operation. H
is a fundamental example of a sub-Riemannian manifold that is not a Riemannian manifold.All geodesics on H are described explicitly as the following. Without loss of generality,
it is sufficient that we consider a geodesic starting at the origin, since the distance function dis left invariant. We identify R
3 by C × R. Any geodesic γϕ : R −→ R3 starting at the origin
(0, 0, 0) with the initial velocity (v, 0) ∈ C × R is described as
γϕ(t) =⎧⎨⎩
(v · e iϕt − 1
i ϕ, |v|2
(ϕ t − sin(ϕ t)
2 ϕ2
) )if ϕ = 0,
(tv, 0) if ϕ = 0,
for a parameter ϕ ∈ R, where the operation · and the absolute value | · | are defined forcomplex numbers. See [20]. We take an arbitrary real number ϕ = 0 and fix it. Put γϕ(t) =(x(t)+ iy(t), z(t)). Then (x(t)+ iy(t), 0) draws a circle in C starting at the origin and z(t)equals the area of the region in C bounded by the arc joining the origin to (x(t)+ iy(t), 0)and the segment joining the origin to (x(t)+ iy(t), 0) as is seen in the following figure.
In this case, γϕ is not a straight line. The z-axis is the cut locus of the origin. In the case ofϕ = 0, γϕ is a line in C passing through the origin in the usual sense and is also a straight linein the sense of the sub-Riemannian distance function d . The general (2n + 1)-dimensionalHeisenberg group H
n for n = 1, 2, . . . is defined by the same way as in the 3-dimensionalcase [20]. H
n is R2n+1 as a manifold. The analogues of facts mentioned above for the 3-
dimensional case also hold for the (2n +1)-dimensional case. There does not exist nontrivial
123
Geom Dedicata (2014) 168:177–196 195
singular minimizers in Hn [24]. According to [20], Hn satisfies MCP(0, 2n+3)with respect to
the Lebesgue measure on R2n+1. From these facts, H
n satisfies the assumptions of Theorems2 and 3. Any straight line γ in H
n induces a diffeomorphism between the product manifoldb−1γ (0) × R and H
n , however there does not exist any metric space X such that X × R andH
n are isometric.
Acknowledgments The author thanks Prof. Takashi Shioya for helpful advices especially giving him anidea and a method to study a splitting theorem.
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