Riemannian convexity of functionals

Constantin Udriste Andreea Bejenaru

Abstract

This paper extends the Riemannian convexity concept to actionfunctionals defined by multiple integrals associated to Lagrangian dif-ferential forms on first order jet bundles. The main results of thispaper are based on the geodesic deformations theory and their impacton functionals in Riemannian setting. They include the basic prop-erties of Riemannian convex functionals, the Riemannian convexityof functionals associated to differential m-forms or to Lagrangians ofclass C1 respectively C2, the generalization to invexity and geometricmeaningful convex functionals.

The Riemannian convexity of functionals is the central ingredientfor global optimization. We illustrate the novel features of this theory,as well as its versatility, by introducing new definitions, theorems andalgorithms that bear upon the currently active subject of functionalsin variational calculus and optimal control. In fact so deep rooted isthe convexity notion that nonconvex problems are tackled by devisingappropriate convex approximations.

Keywords Geodesic deformation; Riemannian convex function-als; Riemannian convex functions; Optimization; Variational prob-lems; Multitime optimal control.

1 Introduction

Convexity is a property of crucial importance in deterministic approaches toglobal optimization. This paper is therefore motivated by the necessity ofcreating a consistent theory related to convex action-type functionals, neededwithin the process of finding complete descriptions for the optimal solutionsof Riemannian multitime variational problems.

0University Polytechnic of Bucharest, Faculty of Applied Sciences, Depart-ment of Mathematics-Informatics I, Splaiul Independentei 313, ROMANIA, e-mail:anet.udri@yahoo.com, bejenaru.andreea@yahoo.com, Tel: 004021 4029150, Fax: 0040213181001

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The research in convex functions registers a boost especially after thesecond World War, mainly for its utility in various domains (economy, tech-nique, physics, chemistry, medicine, sociology etc), since convexity plays afundamental role in the analysis and development of global optimization al-gorithms. The development of the convex analysis was stimulated by theintensive applicative requests: mathematical models and programs of differ-ent types, strategic games, dynamic and optimal control problems that aredescribed in the linear faze with the help of convex functions (and later,generalized convex functions).

From 1964 new insights have been gained in old problems combined withnew ones, and great coherence has been achieved in understanding the role ofRiemannian convexity in science. Now, convex functions occur abundantly(see [5]-[7], [11]-[13], [15], [17], [24]), have many structural implications onmanifolds and form an important link between the modern analysis and ge-ometry. These implications do not occur in the extensive theory of convexfunctions on Euclidean spaces because of the particularity of these spaces.

The main goal of this paper is to pass from Riemannian convexity of func-tions to Riemannian convexity of functionals such that, by customization, weuncover the well-known results for convex functions. The similitude betweenthis paper and the above-quoted books, and also its novelty in the study ofthe convex functionals, consist in the introduction and use of geodesic defor-mations on the domain of the functional, playing the same role as geodesicson the Riemannian domain of a convex function. Since we have brought intodebate the domain of a functional, we insist on the fact that our geodesic de-formations have also the significance of giving to such a domain a geometricmeaning (in general, this set fails in having geometric properties unlike thedomain of a Riemannian convex function, which lays, usually, in a Rieman-nian manifold). In our opinion, this paper focuses on aspects which make ourapproach on convexity particularly attractive from both global optimizationand geometric point of view.

To create the Riemannian convexity of functionals, we put together anddevelop several ideas in Differential Geometry [1]-[3], Variational Calculusand Optimal Control [9], [10], [19]-[22], Riemannian Convexity and Opti-mization [4], [8], [14], [16], [19], [23].

Section 2 describes the Riemannian setting, inspired by multitime vari-ational calculus and defines the geodesic deformations, the totally convexsubsets and the convex functionals. In Section 3, there are given propertiesof the Riemannian convex functionals. The remarkable result of Section 4states the continuity of convex functionals, relative to a particular topologygenerated by the Riemannian structures. Sections 5 and 6 are focused to an-alyzing convexity, when the functionals are associated with C1, respectively

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C2 Lagrangians. The main result consists in giving a description for Rie-mannian convex actions corresponding to C2 Lagrangians, using a Hessianoperator. Section 7 justifies the utility of our approach via the Riemanniankinetic energy as convex functional. Section 8 points out the main outcomes.

2 Geodesic deformations and convex

functionals in Riemannian setting

Let (M, g) be an n-dimensional Riemannian manifold and (N, h) be a com-pactm-dimensional Riemannian manifold (see [1]-[3]). We denote by J1(N,M)the first order jet bundle and by G = h+ g+ h−1⊗ g its induced metric (see[18]), where ⊗ means the tensor product. We suppose that (J1(N,M), G)is a complete Riemannian manifold. Let E be a set of submanifold mapsfrom N to J1(N,M) and E(I) be the set of those submanifold maps thatare integral maps for a family I of differential 1-forms on J1(N,M). If θ isa differential m-form on J1(N,M) we can associate the functional (multipleintegral)

Jθ : E(I)→ R, Jθ[Φ] =

∫N

Φ∗θ, (1)

where Φ∗θ denotes the pull-back of θ on N .For a fixed submanifold map Φ ∈ E(I) we associate a deformation map

ϕ : N × (−δ, δ)→ J1(N,M), satisfying ϕ(·, 0) = Φ. We denote by

XΦ ∈ XΦ(J1(N,M)), X(Φ(t)) = ϕ∗∂

∂ε|ε=0

the infinitesimal deformation induced by ϕ, that is, XΦ is the vector fieldalong Φ(N) satisfying XΦ(t) = ∂ϕ

∂ε(t, 0), ∀t ∈ N . Since all the maps from E(I)

preserve the boundary of N , it follows that XΦ also satisfies the conditionXΦ(t) = 0, ∀t ∈ ∂N . From now on, TΦE(I) will denote the set of allinfinitesimal deformations of Φ as above and

TE(I) = ∪Φ∈E(I)TΦE(I).

It seems natural to consider also the set

X (E(I)) = X ∈ X (J1(N,M))| XΦ = X|Φ(N) ∈ TΦE(I), ∀Φ ∈ E(I). (2)

Our basic example is the multitime variational problem: (N, h) a com-pact m-dimensional Riemannian manifold with local coordinates (t1, ..., tm);(M, g) an n-dimensional manifold with local coordinates (x1, ..., xn); the in-duced local coordinates (tγ, xi, xiγ) on J1(N,M); L : J1(N,M) → R a

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Lagrangian and θ the Cartan form associated with L, that is θ = Ldt +∂L∂xiγ

ωi ∧ dtγ, where I = ωi ∈ Λ1(J1(N,M))| ωi = dxi − xiσdtσ, i = 1,m,dtγ = i ∂

∂tγdt and ∧ stands for exterior product. If we introduce the set

E(I) = Φ : N → J1(N,M)| Φ(t) = (t, x(t),∂x

∂t(t)),

then its tangent space is

TΦE(I) = X = (Xγ, X i, X iγ)| Xγ Φ = 0, Dγ(X

i Φ) = X iγ Φ,

X i(Φ(t)) = 0, ∀t ∈ ∂N(3)

and Jθ : E(I)→ R is defined by Jθ[Φ] =∫N

Φ∗θ =∫NLΦdt = JL[Φ]. From

now on, when dealing with such a variational problem, we replace Jθ by JL,emphasizing the fact that the m-form θ is related to the Lagrangian L.

We return to the general problem and, after associating to E(I) the previ-ous structures on J1(N,M), we look for analyzing their geometric propertiesand carrying these properties back to the set E(I). The next two lemmasprove to be some very useful tools for further development of this theory.

Lemma 1. If M1 and M2 are two differentiable manifolds, T ∈ T 0p (M2)

is a tensor field on M2, Φ : M1 → M2 is a differentiable map and ϕ :M1 × (−δ, δ)→M2 is a deformation of Φ, then

d

dεϕ∗εT |ε=0 = Φ∗(X(T )), (4)

where ϕε : M1 →M2, ϕε(t) = ϕ(t, ε), X = ϕ∗(ddε

) and X(T ) denotes the Liederivative of the tensor field T with respect to the vector field X.

Hint. We evaluate the previous equality in local coordinates. utIn particular, the foregoing result is valid for differential forms, too.Lemma 2. With the same hypotheses as above, if X = ϕ∗

∂∂ε

and Zγ =ϕ∗

∂∂tγ

, then[X,Zγ] Φ = 0, ∀γ ∈ 1,m. (5)

Hint. We use local coordinates. utRemark. The immediate consequence of the previous two results is that

d

dε[ϕ∗εT ]α1...αp |ε=0 = X(T (Zα1 , ..., Zαp)) Φ.

We return now to our basic problem.Lemma 3. If Φ ∈ E(I), then XΦ ∈ TΦE(I) iff

Φ∗(X(ω)) = 0, ∀ω ∈ I, (6)

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where X = ϕ∗∂∂ε

, ϕ being a deformation of Φ induced by XΦ, and X(ω)denotes the Lie derivative.

Proof. Consequence of the relation (4). utRemark. From the definition of the Lie derivative, it follows that the

previous condition can be rewritten as

Φ∗(iXdω) + dΦ∗(iXω) = 0, ∀ω ∈ I, (7)

where iX is the contraction by vector field X.The foregoing Lemmas allow us now to give a geometric interpretation to

E(I). We remark that X (E(I)) is not an F(J1(N,M))-module. Therefore,we consider instead the set X (E(I)) representing the F(J1(N,M))-modulegenerated by X (E(I)) and we also denote by TΦE(I) the F(J1(N,M))-module generated by TΦE(I).

Lemma 4. The set X (E(I)) is an involutive distribution on J1(N,M)and, for each Φ ∈ E(I), there is an integral submanifold AΦ ⊂ J1(N,M)such that XΨ(AΦ) = TΨE(I), ∀Ψ ∈ E(I) a submanifold map resulting aftera deformation of Φ in E(I).

Hint. We verify that X (E(I)) satisfies the definition of an involutivedistribution (see [1]). ut

Remark. If E(I) is connected, then Ψ(N) ⊂ AΦ, ∀Φ,Ψ ∈ E(I). Other-wise, for each connected component of E(I) we can associate a submanifoldas we did above and we consider A to be the union of the previous subman-ifolds. The submanifold A is called the image of E(I) on J1(N,M).

Definition. A deformation map ϕ : N × [0, 1] → J1(N,M) is calledgeodesic deformation if ϕ(t, ·) is a geodesic in (A,G), for each t ∈ N .

Definition. A subset F ⊂ E(I) is called totally convex if, for all pairs ofsubmanifold maps Φ,Ψ ∈ F and each geodesic deformation ϕ : N × [0, 1]→J1(N,M), ϕ(·, 0) = Φ, ϕ(·, 1) = Ψ, we have

ϕ(·, ε) ∈ F, ∀ε ∈ [0, 1]. (8)

Definition. Let F ⊂ E(I) be a totally convex subset of submanifoldmaps and let θ be a differential m-form on J1(N,M). The functional

Jθ : F → IR, Jθ[Φ] =

∫N

Φ∗θ

is called Riemannian convex if

Jθ[ϕ(·, ε)] ≤ (1− ε)Jθ[Φ] + εJθ[Ψ], (9)

for all Φ and Ψ in F , for each geodesic deformation ϕ : N×[0, 1]→ J1(N,M)connecting Φ and Ψ and for all ε ∈ [0, 1].

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The functional Jθ is called Riemannian strictly convex if

Jθ[ϕ(·, ε)] < (1− ε)Jθ[Φ] + εJθ[Ψ], (10)

for all Φ, Ψ, ϕ as above, Φ 6= Ψ and ε ∈ (0, 1).Remark. We can define the convexity of a functional associated with a

differential form outside the variational context. For that we need an arbi-trary Riemannian manifold instead of a jet bundle and the set of the subman-ifold maps between a differential manifold and a Riemannian one. Moreover,we can also consider more general types of functionals, defined using pull-backs of tensor fields instead of differential forms. If so, by customization,we uncover from this theory the Riemannian convexity of functions if takingN = t0, (M, g) a Riemannian manifold and, for E the set of all the mapsfrom N to M and f a differentiable function on M , by considering

Jf : E → R, Jf [Φ] = f Φ.

3 Properties of convex functionals

In this section we prove some elementary properties of convex functionalsfollowing the pattern developed by the theory of convex functions (see [17]).

Theorem 1. The functional Jθ : F → IR is convex (strictly convex) iff,for each geodesic deformation ϕ : N × [0, 1] → J1(N,M) lying in F , thefunction

Jϕ : [0, 1]→ IR, Jϕ(ε) = Jθ[ϕ(·, ε)] (11)

is convex (strictly convex) on the interval [0, 1].Proof. Let us suppose that ϕ : N × [0, 1] → J1(N,M) is a geodesic

deformation on F and the associated function Jϕ is convex on [0, 1]. Itfollows that

Jϕ(ε) ≤ (1− ε)Jϕ(a) + εJϕ(b), ∀[a, b] ⊂ [0, 1], ∀ε ∈ [0, 1].

In particular, for a = 0 and b = 1, we obtain Jϕ(ε) ≤ (1−ε)Jϕ(0)+εJϕ(1),that is

Jθ[ϕ(·, ε)] ≤ (1− ε)Jθ[ϕ(·, 0)] + εJθ[ϕ(·, 1)],

and, since ϕ is arbitrary, it follows that Jθ is convex.Let us prove now the converse. We suppose that Jθ is convex and we

consider ϕ : N × [0, 1] → J1(N,M) a geodesic deformation on F and Jϕthe associated function. If [a, b] ⊂ [0, 1], we can associate another geodesicdeformation defined by

ψ : N × [0, 1]→ J1(N,M), ψ(t, ε) = ϕ(t, (1− ε)a+ εb). (12)

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It follows that ψ(t, 0) = ϕ(t, a) and ψ(t, 1) = ϕ(t, b). Since Jθ is convex,the following inequality holds

Jθ[ψ(·, ε)] ≤ (1− ε)Jθ[ψ(·, 0)] + εJθ[ψ(·, 1)],

which is equivalent to

Jθ[ϕ(·, (1− ε)a+ εb)] ≤ (1− ε)Jθ[ϕ(·, a)] + εJθ[ϕ(·, b)],

that is Jϕ((1− ε)a+ εb) ≤ (1− ε)Jϕ(a) + εJϕ(b). Since a, b were arbitrary in[0,1], it follows that Jϕ is convex on this interval.

The strict convexity is proven similarly. utTheorem 2. If ϕ : N × [0, 1] → J1(N,M) is a geodesic deformation

connecting Φ,Ψ ∈ F , and ϕ is a geodesic extension of ϕ in F , then

1. Jθ : F → IR is a convex functional iff

Jθ[ϕ(·, ε)] ≥ (1− ε)Jθ[Φ] + εJθ[Ψ], (13)

for all ε ≤ 0, for all Φ,Ψ ∈ F and for each extended geodesic deforma-tion ϕ passing through Φ and Ψ.

2. Jθ : F → IR is a convex functional iff

Jθ[ϕ(·, ε)] ≥ (1− ε)Jθ[Φ] + εJθ[Ψ], (14)

for all ε ≥ 1, for all Φ,Ψ ∈ F and each extended geodesic deformationϕ passing through Φ and Ψ.

Hint. See [17] for an identical proof of the equivalent result related toRiemannian convex functions. ut

Corollary 1. If a functional Jθ : E(I)→ IR is convex and upper bounded,then Jθ is constant. Moreover, if JL is a convex and bounded action associatedwith a multitime variational problem, then L is a null Lagrangian.

Hint. See [17], [19]. utTheorem 3. Let F ⊂ E(I) be a totally convex subset and Jθ : F → IR be

a convex functional. If I is the F(J1(N,M))-module generated by I and iff : J1(N,M)→ J1(N,M) is a diffeomorphism preserving I, that is f ∗(I) =I, then f(F ) = f Φ| Φ ∈ F is a totally convex subset and J(f−1)∗θ is aconvex functional on f(F ).

Proof. If Φ ∈ F , then f Φ ∈ f(F ). Since f ∗ω ∈ I, ∀ω ∈ I, it followsΦ∗(f ∗ω) = (f Φ)∗ω = 0, ∀ω ∈ I, proving that f Φ ∈ E(I). Thereforef(F ) ⊂ E(I).

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Let ϕ : f(F )× [0, 1]→ J1(N,M) be a geodesic deformation between f Φand f Ψ, where Φ,Ψ ∈ F . We know that, for a fixed point t ∈ N , the curveε → ϕ(t, ε) is a geodesic on J1(N,M) and, since f is a diffeomorphism onJ1(N,M), it follows that ε → (f−1 ϕ)(t, ε) is also a geodesic. Therefore(f−1ϕ) is a geodesic deformation between Φ and Ψ. That is, (f−1ϕ)(·, ε) ∈F, ∀ε ∈ [0, 1] and it follows that ϕ(·, ε) ∈ f(F ), ∀ε ∈ [0, 1] and f(F ) ⊂ E(I)is a totally convex subset. Moreover, we have

J(f−1)∗θ[ϕ(·, ε)] =

∫N

ϕ∗ε((f−1)∗θ) =

∫N

(f−1 ϕε)∗θ

= Jθ[f−1 ϕ(·, ε)] ≤ (1− ε)Jθ[Φ] + εJθ[Ψ]

= (1− ε)J(f−1)∗θ[f Φ] + εJ(f−1)∗θ[f Ψ].

Therefore, J(f−1)∗θ is a convex functional. utCorollary 2. Let JL be a convex action associated with a multitime vari-

ational problem. If f : J1(N,M)→ J1(N,M) is a diffeomorphism preservingI, then JLf−1 is also a convex functional.

Proof. Since f ∗(I) = I, we have previously proved that f(E(I)) ⊂ E(I)is a totally convex subset. Let θ be the Cartan form associated with L andΨ ∈ f(E(I)), that is Ψ = f Φ, with Φ ∈ E(I). Then Ψ∗((f−1)∗θ) =Φ∗θ = LΦdt = (Lf−1)Ψdt, therefore J(f−1)∗θ = JLf−1 and, by applyingTheorem 3, we conclude that JLf−1 is a convex functional. ut

Corollary 3. If JL : E(I) → IR is a functional as above, then theproperty of JL of being convex is invariant under any change of coordinateson M .

Proof. We consider (x1, .., xn) and (x1, ..., xn), two systems of coordinateson M , and (t1, ..., tm) some local coordinates on N . They generate two setsof coordinates (t, xi, xiγ) and (t, xi, xiγ) on J1(N,M). If f : J1(N,M) →J1(N,M) is the diffeomorphism associated with this change of coordinates,then f ∗(I) = I. Indeed,

f ∗ωi = f ∗(dxi − xiσdtσ) =∂xi

∂xjdxj − ∂xi

∂xjxjσdt

σ =∂xi

∂xjωj ∈ I.

It follows f ∗(I) ⊂ I, which proves that f ∗(I) ⊆ I. Since f : J1(N,M) →J1(N,M) is a diffeomorphism, we have f ∗(I) = I. If L = L(t, xi, xiγ) =L f−1(t, xi, xiγ), then, by applying the foregoing Corollary, we obtain: if JLis convex, then JLf−1 is also convex. ut

Theorem 4. Let F ⊂ E(I) be a totally convex subset and Jθ : F → IRbe a convex functional. If Im(F ) ⊂ (a, b) and (a, b) ⊂ IR is an open interval,and if f : (a, b)→ IR is a convex increasing function, then f Jθ is a convexfunctional on F .

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Proof. Let Φ, Ψ ∈ F and ϕ : N × [0, 1] → J1(N,M) be a geodesicdeformation between Φ and Ψ on F . Then

f(Jθ[ϕ(·, ε)]) ≤ f ((1− ε)Jθ[Φ] + εJθ[Ψ]) ≤ (1− ε)f(Jθ[Φ]) + εf(Jθ[Ψ]),

which proves that f Jθ is a convex functional on F . utTheorem 5. If F ⊂ E(I) is a totally convex subset, Jθa : F → IR, a =

1, k are convex functionals and ca ∈ IR, ca ≥ 0, ∀a = 1, k, then caJθa andsupa=1,k Jθa are convex functionals.

Proof. For a geodesic deformation ϕ : N × [0, 1] → J1(N,M) betweentwo submanifold maps Φ, Ψ ∈ F , we have

(caJθa)[ϕ(·, ε)] = caJθa [ϕ(·, ε)] ≤ ca[(1− ε)Jθa [Φ] + εJθa [Ψ]]

= (1− ε) (caJθa) [Φ] + ε (caJθa) [Ψ]

and

( supa=1,k

Jθa)[ϕ(·, ε)] = supa=1,k

Jθa [ϕ(·, ε)] ≤ supa=1,k

[(1− ε)Jθa [Φ] + εJθa [Ψ]]

= (1− ε)

(supa=1,k

Jθa

)[Φ] + ε

(supa=1,k

Jθa

)[Ψ].

utTheorem 6. Let Jθ : F → IR be a convex functional, where F ⊂ E(I) is

a totally convex subset and c ∈ IR. Then F c = Φ ∈ F | Jθ[Φ] ≤ c ⊂ F is atotally convex subset.

Proof. If Φ and Ψ are two submanifold maps in F c and ϕ : N × [0, 1]→J1(N,M) is a geodesic deformation between them, then

Jθ[ϕ(·, ε)] ≤ (1− ε)Jθ[Φ] + εJθ[Ψ] = (1− ε)c+ εc = c, ∀ε ∈ [0, 1],

therefore ϕ(·, ε) ∈ F c, ∀ε ∈ [0, 1] and F c is a totally convex subset. utTheorem 7. If Φ ∈ E(I) is a local minimum point for a convex func-

tional Jθ : E(I)→ IR, then Φ is a global minimum point.Proof. Let us suppose that Φ is a local minimum point for a convex

functional Jθ. It follows that there is a neighborhood F ⊂ E(I) of Φ suchthat Jθ[Φ] ≤ Jθ[Ψ], ∀Ψ ∈ F . Let ξ be an arbitrary submanifold map inE(I), ϕ : N × [0, 1]→ J1(N,M) be a deformation between Φ and ξ and letδ ∈ (0, 1) such that ϕ(·, δ) ∈ F . If we denote Ψ = ϕ(·, δ), it follows

Jθ[ϕ(·, ε)] ≤ (1− ε)Jθ[Φ] + εJθ[ξ] ≤ (1− ε)Jθ[Ψ] + εJθ[ξ]

= (1− ε)Jθ[ϕ(·, δ)] + εJθ[ξ]

≤ (1− ε)(1− δ)Jθ[Φ] + (1− ε)δJθ[ξ] + εJθ[ξ].

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Replacing ε = 0, we find

Jθ[Φ] ≤ (1− δ)Jθ[Φ] + δJθ[ξ]⇒ δJθ[Φ] ≤ δJθ[ξ]⇒

Jθ[Φ] ≤ Jθ[ξ].

Since ξ ∈ E(I) was arbitrary, it follows that Φ is a global minimum point.ut

Theorem 8. The set Argmin (Jθ), consisting in all the submanifold mapsfrom E(I) that minimize the convex functional Jθ, is a totally convex subset.

Proof. Let Φ,Ψ ∈ Argmin (Jθ) and ϕ : N × [0, 1] → J1(N,M) be ageodesic deformation connecting them. Since Jθ is convex, it follows

Jθ[ϕ(·, ε)] ≤ (1− ε)Jθ[Φ] + εJθ[Ψ] ≤ (1− ε)Jθ[ξ] + εJθ[ξ] = Jθ[ξ], ∀ξ ∈ E(I).

The previous inequality states the fact that ϕ(·, ε) is a minimum pointfor Jθ, therefore ϕ(·, ε) ∈ Argmin (Jθ), ∀ε ∈ [0, 1]. ut

Definition. Let Jθ : E(I) → IR be the functional associated with thedifferential m-form θ. The map

dJθ(Φ) : TΦE(I)→ IR, dJθ(Φ)[XΦ] =d

dεJ [ϕ(·, ε)]|ε=0, (15)

where ϕ : N × (−δ, δ) → J1(N,M) is a deformation of Φ in E(I) such thatXΦ(t) = ∂ϕ

∂ε(t, 0), is called the differential of the functional Jθ at Φ.

Remark. The differential of the functional Jθ at Φ can be also expressedby

dJθ(Φ)[XΦ] = JX(θ)[Φ] =

∫N

Φ∗(X(θ)) =

∫N

Φ∗(iXΦdθ), ∀XΦ ∈ TΦE(I),

(16)where X is the vector field along a deformation of Φ induced by XΦ.

Definition. A map Φ ∈ E(I) is called critical point of the functional Jθif dJθ(Φ)[XΦ] = 0, ∀XΦ ∈ TΦE(I).

Theorem 9. Let F ⊂ E(I) be a totally convex subset and Jθ : F → IRbe a convex functional. If Φ ∈ F is a critical point for Jθ, then Φ minimizesthe functional.

Proof. We know that Φ ∈ F is a critical point for Jθ iff ddεJθ[ϕ(·, ε)] |ε=0=

0, ∀ϕ : N × [0, 1]→ J1(N,M) a deformation of Φ on F , therefore iff ε = 0 isa critical point for the function Jϕ(ε) = Jθ[ϕ(·, ε)]. On the other side, since Jθis convex, it follows that Jϕ is convex on [0, 1], therefore ε = 0 is a minimumpoint for Jϕ, for any deformation ϕ, so Φ minimizes the functional Jθ. ut

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4 The continuity of a convex functional

This section establishes the continuity of convex functionals, relative to aproper topology induced by the Riemannian structures involved. Let Φ andΨ be two submanifold maps in E(I) and

ΓΦΨ = ϕ : N×[0, 1]→ J1(N,M)| ϕ(·, 0) = Φ; ϕ(·, 0) = Ψ; ϕ(·, ε) ∈ E(I).

We define

d : E(I)× E(I)→ R+, d(Φ,Ψ) = infϕ∈ΓΦΨ

(∫N

∫ 1

0

||∂ϕ∂ε

(t, ε)||dεdt), (17)

where the norm above is related to the Riemannian metric G on the jetbundle J1(N,M).

Remark. If dG : J1(N,M) × J1(N,M) → R+ is the distance functionon J1(N,M) induced by the Riemannian metric G, then

d1(Φ,Ψ) =

∫N

dG(Φ(t),Ψ(t))dt ≤ d(Φ,Ψ), ∀Φ,Ψ ∈ E(I).

The function d defined by the relation (17) is a distance on the domainE(I) and generates a topology on E(I). We define next the continuity of afunctional relative to this topology.

Definition. A functional Jθ : E(I) → R is called continuous at Φ if,∀ε > 0, there is δ > 0 such that

|Jθ[Ψ]− Jθ[Φ]| < ε, ∀Ψ ∈ E(I), d(Φ,Ψ) < δ.

Theorem 10. Let F ⊂ E(I) be an open totally convex subset and letJθ : F → R be a convex functional. Then Jθ is continuous on F , relative tothe topology generated by the distance d.

Proof. Let Φ0 ∈ E(I), c ∈ R and r > 0 such that B(Φ0, r) ⊂ F c,where B(Φ, r) denotes the closed ball of radius r in E(I), centered at Φ.Furthermore, let ϕ : N × [−1, 1] → J1(N,M) be a geodesic deformation inB(Φ0, r), such that ϕ(·, 0) = Φ0, ϕ(·, 1) = Φ1, ϕ(·,−1) = Φ2 and Φ1,Φ2 ∈∂B(Φ0, r). The definition of d allows us to consider a parametrization of ϕ,

proportional with the distance, i.e., if ϕ(·, δ) = Φ, then δ = d(Φ0,Φ)r

.Since Jθ is a convex functional, it follows

Jθ[ϕ(·, ε)] ≤ (1− ε)Jθ[Φ0] + εJθ[Φ1] ≤ (1− ε)Jθ[Φ0] + εc,∀ε ∈ [0, 1].

If δ ∈ [0, 1], let ϕ(·, δ) = Φ. Then

Jθ[Φ]− Jθ[Φ0] ≤ δ(c− Jθ[Φ0]). (18)

11

If ϕ|[−1,δ] is the geodesic deformation between Φ2 and Φ, we denoteα(·, s) = ϕ(·,−1 + s(δ + 1)), s ∈ [0, 1]. Then α(·, 0) = Φ2 and α(·, 1) = Φ,and, since Jθ is convex, we have

Jθ[α(·, s)] ≤ (1− s)Jθ[Φ2] + sJθ[Φ] ≤ (1− s)c+ sJθ[Φ],∀s ∈ [0, 1].

When taking s = 11+δ

, we obtain Jθ[Φ0] ≤ (1− 11+δ

)c+ 11+δ

Jθ[Φ], that is

Jθ[Φ]− Jθ[Φ0] ≥ δ(Jθ[Φ0]− c). (19)

Combining (18) and (19), we conclude that

|Jθ[Φ]− Jθ[Φ0]| ≤ δ(c− Jθ[Φ0]) =(c− Jθ[Φ0])

rd(Φ,Φ0).

The previous inequality proves the continuity of Jθ. utRemark. The choice of the distance d plays a crucial role here. In order

to prove this statement, we invoke the following example:

E = x : [0, 1]→ R+| x(·) continuous

and the ”negative” Boltzmann-Shannon entropy functional

J : E → R, J [x(·)] =

∫ 1

0

x(t) ln(x(t))dt.

This functional is convex and the foregoing Theorem proves that it is also con-tinuous on the metric space (E, d), where d(x(·), y(·)) = infϕ

∫ 1

0

∫ 1

0|∂ϕ∂ε

(t, ε)|dεdt,ϕ denoting a deformation between x(·) and y(·). Nevertheless, a remark in[4] classifies the previous functional as everywhere discontinuous (but lowersemicontinuous) on L1[0, 1].

5 Convexity for C1 Lagrangians

In [17], some interesting properties of Riemannian convex C1 functions areproved. We give next some equivalent descriptions for convex functionalsassociated with differentiable forms or with C1 Lagrangians.

Theorem 11. Let F ⊂ E(I) be an open totally convex subset and Jθ :F → IR be a functional associated with a differential m-form θ on J1(N,M).The functional Jθ is convex iff, for each geodesic deformation ϕ : N×[0, 1]→J1(N,M), the function Jϕ : [0, 1] → IR, Jϕ(ε) = Jθ[ϕ(·, ε)] satisfies thecondition

Jϕ(ε)− Jϕ(δ) ≥ (ε− δ)J ′ϕ(δ), ∀ε, δ ∈ [0, 1]. (20)

12

The functional Jθ is strictly convex iff

Jϕ(ε)− Jϕ(δ) > (ε− δ)J ′ϕ(δ), ∀ε, δ ∈ (0, 1), ε 6= δ. (21)

Hint. We use the equivalence between the convexity of the functional Jθand the convexity of the function Jϕ. ut

Theorem 12. The functional Jθ : F → IR is convex iff

Jθ[Ψ]− Jθ[Φ] ≥ dJθ(Φ)[X], ∀Φ,Ψ ∈ F, (22)

where X ∈ TΦE(I) is the infinitesimal deformation associated with a geodesicdeformation between Φ and Ψ.

Moreover, the functional Jθ : F → IR is strictly convex iff

Jθ[Ψ]− Jθ[Φ] > dJθ(Φ)[X], ∀Φ 6= Ψ ∈ F. (23)

Proof. We suppose that Jθ is convex. Then, for each pair of submanifoldmaps (Ψ,Φ) in F and each geodesic deformation ϕ : N × [0, 1]→ J1(N,M)between them, we have

Jϕ(ε)− Jϕ(δ) ≥ (ε− δ)J ′ϕ(δ), ∀ε, δ ∈ [0, 1].

In particular, for ε = 1 and δ = 0, we obtain

Jθ[Ψ]− Jθ[Φ] ≥ d

dεJθ[ϕ(·, ε)]|ε=0 = dJθ(Φ)[X],

where X is the infinitesimal deformation associated with ϕ.Let us prove now the converse of this statement. We consider Φ,Ψ ∈ F

and ϕ a geodesic deformation between them. For δ, ε ∈ [0, 1], we introducethe map

ψ : N × [0, 1]→ J1(N,M), ψ(t, s) = ϕ(t, (1− s)δ + sε).

Then ψ is a geodesic deformation between ϕ(·, δ) and ϕ(·, ε) and

Jψ(s) = Jθ[ψ(·, s)] = Jθ[ϕ(·, (1− s)δ + sε)] = Jϕ((1− s)δ + sε).

The relation

Jθ[ψ(·, 1)]− Jθ[ψ(·, 0)] ≥ JY (θ)[ψ(·, 0)],

13

where Y is the tangent vector field associated with the deformation ψ, isequivalent to

Jψ(1)− Jψ(0) ≥ J ′ψ(0),

orJϕ(ε)− Jϕ(δ) ≥ (ε− δ)J ′ϕ(δ).

Since Φ, Ψ and ϕ were arbitrary fixed, the foregoing Theorem ensures usthat Jθ is convex. ut

Corollary 4. If L is a C1 Lagrangian in a multitime variational problem,then JL is convex iff∫

N

L Ψdt−∫N

L Φdt ≥∫N

X(L) Φdt, ∀Φ,Ψ ∈ F, (24)

where X ∈ TΦE(I) is associated again to a geodesic deformation between Φand Ψ.

Proof. We have JL[Φ] =∫NL Φdt and dJθ(Φ)[X] =

∫NX(L) Φdt. ut

Definition. Let Jθ : F → IR be the functional associated with an m-formθ. If the differential dJθ satisfies the condition

dJθ(Ψ)[X]− dJθ(Φ)[X] ≥ 0, ∀Φ,Ψ ∈ F,

for each vector field X associated with a geodesic deformation between Φand Ψ, then dJθ is called increasing.

Corollary 5. The functional Jθ is convex on F iff the differential dJθ isan increasing map.

Hint. If ϕ : N × [0, 1]→ J1(N,M) is a geodesic deformation between Φand Ψ, then ψ : N × [0, 1] → J1(N,M), ψ(t, ε) = ϕ(t, 1 − ε) is a geodesicdeformation between Ψ and Φ. ut

Corollary 6. If L is a C1 Lagrangian associated with a multitime vari-ational problem, then JL is convex iff∫

N

dL(X)(Ψ(t))dt ≥∫N

dL(X)(Φ(t))dt, ∀Φ,Ψ ∈ E(I),

for each vector field X ∈ X (E(I)) tangent to a geodesic deformation betweenΦ and Ψ.

Hint. Same arguments as in the proof of Corollary 4. ut

6 Convexity for C2 Lagrangians

Theorem 13. Let F ⊂ E(I) be an open totally convex subset and Jθ : F →IR be the functional associated with a differential m-form θ. The functional

14

Jθ is convex iff, for each geodesic deformation ϕ : N × [0, 1] → J1(N,M),the function Jϕ : [0, 1]→ IR, Jϕ(ε) = Jθ[ϕ(·, ε)] satisfies the condition

d2Jϕdε2

≥ 0, ∀ε ∈ [0, 1]. (25)

Ifd2Jϕdε2

> 0, ∀ε ∈ [0, 1], (26)

then Jθ is strictly convex.Hint. Same as for Theorem 11. utTheorem 14. The functional Jθ is convex iff

JX(X(θ))[Φ] =

∫N

Φ∗(X(X(θ))) ≥ 0, ∀Φ ∈ E(I), ∀X ∈ X (E(I)).

IfJX(X(θ))[Φ] > 0, ∀Φ ∈ E(I), ∀X 6= 0 ∈ X (E(I)),

then Jθ is strictly convex.Hint. See Theorem 12 and the properties of convex C2 functions ([17]).

utLet us consider Φ ∈ F a submanifold map and ϕ : N × [0, 1]→ J1(N,M)

a geodesic deformation of Φ in F . We denote Xϕ(t,ε) = ∂ϕ∂ε

(t, ε) the vector fieldtangent to this geodesic deformation. Since ε→ ϕ(t, ε) is a geodesic in A, itfollows that (∇A

XX)(ϕ(t, ε)) = 0, ∀t ∈ N, ∀ε ∈ [0, 1]. This is precisely themoment that reflects the necessity of working with geodesic deformations.

Theorem 15. Let L be a C2 Lagrangian associated with a multitimevariational problem. If the bilinear form HessAL on J1(N,M) is positive[definite] semidefinite on TΦE(I), ∀Φ ∈ E(I), then JL is [strictly] convex.

Proof. By computation, we have

d2ϕαdε2

=d2

dε2

∫N

L α(t, ε)dt =

∫N

X(X(L))(α(t, ε))dt

=

∫N

HessAL(X,X)(α(t, ε))dt.

Therefore, if

HessAL(X,X) Φ ≥ 0, ∀Φ ∈ F, ∀X ∈ TΦE(I),

then the functional JL is convex. Moreover, if HessAL(X,X) Φ > 0, thenthe functional JL is strictly convex. ut

15

Let h : X (A) × X (A) → X (J1(N,M)) denote the second fundamentalform of the submanifold A.

Corollary 7. If L is a C2 Lagrangian associated with a multitime vari-ational problem and the bilinear form h + HessL on J1(N,M) is positive[definite] semidefinite on TΦE(I), ∀Φ ∈ E(I), then JL is [strictly] convex.

Proof. Since A is a submanifold of J1(N,M), we have (the Gauss for-mula)

∇XY = ∇AXY + h(X, Y ), ∀X, Y ∈ X (A).

We obtain

HessAL(X, Y ) = X(Y (L))−∇A(X, Y )(L)

= X(Y (L))−∇(X, Y )(L) + h(X, Y )(L)

= HessL(X, Y ) + h(X, Y )(L)

and we apply Theorem 15. utCorollary 8. We consider two Riemannian manifolds: IRn and the

parallelepiped Ωt0,t1 in IRm, with the Euclidean metric structures, and Lthe Lagrangian associated with a variational problem relative to this twospaces (i.e. an Euclidean variational problem). If the bilinear form HessLon J1(Ωt0,t1 , IR

n) is positive [definite] semidefinite on TΦE(I), ∀Φ ∈ E(I),then the functional JL is [strictly] convex and all the solutions of the Euler-Lagrange PDEs are optimal solutions.

Hint. When we substitute the Riemannian structures with Euclideanones, the second fundamental form of the submanifold A equals zero. ut

7 Riemannian kinetic energy as

convex functional

This section has two main goals: (1) to give an example of a geometricmeaningful convex functional and (2) to suggest the possibility of creatingan extended theory, following the pattern described in this paper, related tothe convexity of more general functionals defined as multiple integrals, whoseintegrands are defined using the pullback operator.

If (N, h) is a closed m-dimensional Riemannian manifold and (M, g) isan n-dimensional Riemannian manifold, let E be the set of all differentiablemaps from N to M .

Definition. Let x(·) ∈ E. The multiple integral

J [x(·)] =1

2

∫N

Trh(x∗g)(t)√h(t)dt

16

is called the kinetic energy of the submanifold map x(·), where x∗g is theRiemannian metric induced by g and the submanifold map x(·) on N .

Although the integrand here is not written as the pullback of some dif-ferential form, we remark that the resemblance of this functional with thefunctional associated with an m-form as in the foregoing sections (i.e. themain part consists in the pull-back of a tensor field), allows us to apply asimilarly technique: to consider geodesic deformations, to differentiate alongthis deformations and to obtain Lie derivatives.

Let ϕ : N × [0, 1]→M be a geodesic deformation of x(·) and X = ϕ∗(∂∂ε

)be the vector field tangent to this geodesic deformation. Moreover, we alsoconsider the family of vector fields Zα = ϕ∗(

∂∂tα

).Lemma 5. If the bilinear form

Ωx(·) = hαβHess(g(Zα, Zβ)) (27)

is positive semidefinite on Xx(·)(M), then the kinetic energy functional J isRiemannian convex at x(·). If Ωx(·) is positive definite, then J is strictlyconvex at x(·).

Proof. We consider the function Jϕ : [0, 1]→ IR, Jϕ(ε) = J [ϕ(·, ε)]. The

kinetic energy J is Riemannian convex at x(·) iff d2Jϕdε2|ε=0 ≥ 0. Furthermore,

if d2Jϕdε2|ε=0 > 0, then J is strictly convex at x(·). By direct computation we

obtain

d2ϕ

dε2(0) =

1

2

∫N

hαβ[X(X(g(Zα, Zβ)))]dt =1

2

∫N

hαβHess(g(Zα, Zβ))dt.

utTheorem 16. If the curvature tensor field RX ∈ T 0

2 (M), RX(U, V ) =R(X,U,X, V ) is negative semidefinite, ∀X ∈ X (M), then the kinetic energyfunctional is convex.

Proof. For x(·) ∈ E, we consider the family of vector fields Zα = ϕ∗∂∂tα

,with ϕ a geodesic deformation of x(·). If X ∈ X (M) is the vector fieldtangent to this geodesic deformation, then

Hess(g(Zα, Zβ))(X,X) = X(X(g(Zα, Zβ))) = g(∇X∇XZα, Zβ)

+2g(∇XZα,∇XZβ) + g(Zα,∇X∇XZβ).

On the other hand, since [X,Zα] = 0, ∀α ∈ 1,m and ∇XX = 0, it follows

g(∇X∇XZα, Zβ) = −R(Zβ, X, Zα, X),

hence

Hess(g(Zα, Zβ))(X,X) = −2R(Zβ, X, Zα, X) + 2g(∇XZα,∇XZβ).

17

We also have

hαβR(Zβ, X, Zα, X) x = Trh(x∗RX) ≤ 0

and

hαβg(∇XZα,∇XZβ) x = hαβg(∇ZαX,∇ZβX) x = Trh(x∗ΩX),

where ΩX = g(∇X,∇X), and it follows

Ωx(·)(X,X) x = −2Trh(x∗RX) + 2Trh(x∗ΩX).

Proving the fact that ΩX is positive semidefinite, ∀X ∈ X (M) is the laststep. Indeed,

ΩX(Y, Y ) = g(∇YX,∇YX) = ‖∇YX‖2 ≥ 0

and, since RX ≤ 0, we conclude that Ωx(·) is positive semidefinite along x(·)and J is a convex functional. ut

8 Conclusions

1. Our results prove a strong correlation between the convex nature of anaction associated with a Lagrangian and the convex nature of the La-grangian itself. This explains why, for example, the role of the Hessianin the study of the Riemannian convexity of C2 functions is transferredto actions associated with C2 Lagrangians.

2. The study of convex functionals is motivated by the fact that convexvariational problems have the advantage of precisely identifying theoptimal solutions: all the solutions of the Euler-Lagrange PDEs aresolutions for the variational problem.

3. The convexity of a certain functional can be studied independent ofthe variational context by substituting the jet bundle with an arbitrarycomplete Riemannian manifold and by eliminating any restriction re-lated to the domain. Also, the convexity of more general functionals,defined using pull-backs of tensor fields, can be analyzed by followingthe same pattern.

4. An important result related to the continuity of convex functionalsobtained here proves the importance of choosing the proper topology onthe domain of functionals (the topology generated by the Riemannianstructure).

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References

[1] Aubin T.: A Course in Differential Geometry. Amer. Math. Soc., Grad-uate Stud. in Math. (2000).

[2] Bishop R.L., O’Neill B.: Manifolds of negative curvature. Trans. Amer.Mat. Soc. 145, 1-49 (1969).

[3] Boothby W. M.: An Introduction to Differentiable Manifolds and Rie-mannian Geometry. Academic Press (1975).

[4] Borwein J. M., Lewis A. S.: Convex Analysis and Nonlinear Optimiza-tion, Theory and Examples. Springer (2006).

[5] Gordon W.B.: Convex functions and harmonic maps. Proc. Amer. Math.Soc. 33, 433-437 (1972).

[6] Greene R. E., Shiohama K.: Riemannian manifolds having a nowhereconstant convex function. Notices Amer. Math. Soc. 26, 2-a - 223 (1979).

[7] Greene R. E., Wu H.: C∞ Convex functions and manifolds of positivecurvature. Acta. Math. 137, 209-245 (1976).

[8] Hadjisawas N., Pardalos P. M.: Advances in Convex Analysis and GlobalOptimization; honoring the memory of C. Caratheodory (1873-1950).Kluwer Academic Publishers (2001).

[9] Hermann R.: Geometric structure of Systems-Control Theory andPhysics. Interdisciplinary Mathematics, part. A, vol. IX, Math. Sci. Press,Brookline (1974).

[10] Hermann R.: Differential Geometry and the Calculus of Variations. In-terdisciplinary Mathematics, vol. XVII, Math. Sci. Press, Brookline (1977).

[11] Hermann R.: Convexity and pseudoconvexity for complex manifolds.Journal of Math. and Mech. 13, 667-672; 1065-1070 (1964).

[12] Mititelu S.: Generalized Convexities. Geometry Balkan Press (in Roma-nian), Bucharest (2009).

[13] Mititelu S.: Generalized invexity and vector optimization on differen-tiable manifolds. Differential Geometry-Dynamical Systems, 3, No. 1, 21-31 (2001).

[14] Oprea T.: Problems of Constrained Extremum in Riemannian Geome-try. Editorial House of University of Bucharest (in Romanian) (2006).

19

[15] Pop F.: Convex functions with respect to a distribution on Riemannianmanifolds. An. st. Univ. Al. I. Cuza, Iasi, 25, 411-416 (1979).

[16] Rapcsak T.: Smooth nonlinear optimization in Rn. Kluwer AcademicPublishers (1997).

[17] Udriste C.: Convex Functions and Optimization Methods on Rieman-nian Manifolds. Mathematics and Its Applications 297, Kluwer AcademicPublishers, Dordrecht, Boston, London (1994).

[18] Udriste C.: Non-Classical Lagrangian Dynamics and Potential Maps.WSEAS Transactions on Mathematics 7, 12-18 (2008).

[19] Udriste C., Dogaru O., Tevy I.: Null Lagrangian forms and Euler-Lagrange PDEs. J. Adv. Math. Studies, Vol. 1, No. 1-2, 143-156 (2008).

[20] Udriste C., Tevy I.: Multitime Dynamic Programming for Multiple In-tegral Actions. Journal of Global Optimization, DOI: 10.1007/s10898-010-9599-4 (2010).

[21] Udriste C., Tevy I.: Multitime Dynamic Programming for CurvilinearIntegral Actions. Journal of Optimization Theory and Applications, 146,189-207 (2010).

[22] Udriste C.: Simplified multitime maximum principle. Balkan Journal ofGeometry and its Applications, 14, No. 1, 102-119 (2009).

[23] Udriste C., Sandru O., Nitescu C.: Convex programming on thePoincare plane. Tensor N. S. 51, No. 2, 103-116 (1992).

[24] Yau S.T.: Non-existence of continuous convex functions on certain Rie-mannian manifolds. Math. Ann. 207, 269-270 (1974).

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