Convexity and some geometric properties
aa r X i v : . [ m a t h . DG ] D ec Convexity and some geometric properties
J. X. Cruz Neto , ´Italo Melo and Paulo Sousa Abstract.
The main goal of this paper is to present results of existence andnon-existence of convex functions on Riemannian manifolds and, in the caseof the existence, we associate such functions to the geometry of the manifold.Precisely, we prove that the conservativity of the geodesic flow on a Rieman-nain manifold with infinite volume is an obstruction to the existence of convexfunctions. Next, we present a geometric condition that ensures the existence of(strictly) convex functions on a particular class of complete non-compact man-ifolds, and, we use this fact to construct a manifold whose sectional curvatureassumes any real value greater than a negative constant and admits a strictlyconvex function. In the last result we relate the geometry of a Riemannianmanifold of positive sectional curvature with the set of minimum points of aconvex function defined on the manifold.
Key words : Convex function, Geodesic flow, Conformal fields, Soul of a man-ifold.Universidade Federal do Piau´ı, Departamento de Matem´atica, 64049-550,Ininga - Teresina - PI, Brazil email: [email protected] email: [email protected] email: [email protected]
1. Introduction
The concept of convexity plays a very important role in optimization theory,firstly because many objective functions are convex in a sufficiently small neigh-borhood of a local minimum point, and secondly because one can establish theconvergence of numerical methods to estimate minimum points for convex func-tions.Convex functions occur abundantly, with many structural implications on Rie-mannian manifolds and form an important link between modern analysis and ge-ometry. The existence of convex functions implies restrictions on the geometry ortopology of a complete non-compact Riemannian manifold. For example: Bishopand O’Neill [ ] proved that there is no non-trivial smooth convex function on acomplete Riemannian manifold with finite volume; later, Yau [ ] generalized theresult of Bishop and O’Neill, proving that there is no non-trivial continuous convexfunction on a complete manifold with finite volume; Shiohama [ ] proved a result The first author is supported by CNPq/Brazil. The third author is supported by CNPq/Braziland PROCAD/CAPES/Brazil. , ´ITALO MELO AND PAULO SOUSA relating the existence of strictly convex functions to the topology of the Riemann-ian manifold, actually he proved that if a complete Riemannian manifold admits astrictly convex function, then the manifold has at most two ends.In this paper we obtain a geometric-topological restriction to the existence ofnon-trivial convex functions on complete non-compact Riemannian manifolds, morespecifically, we prove that the conservativity of the geodesic flow on a Riemannianmanifold with infinite volume implies that all convex functions on the manifold areconstant, this fact generalizes in a certain sense the result proved by Yau [ ].The technique used in the proof of convergence of many methods of minimiz-ing convex functions on Riemannian manifolds are well-posed when the sectionalcurvature does not change sign. The works of Cruz Neto et al. [ ] and Ferreira-Oliveira [ ] are pioneers in the class of works involving methods of minimizingconvex functions on Riemannian manifolds of non-negative sectional curvature. In2002, Ferreira and Oliveira [ ] established proximal point method in Hadamardmanifolds (non positive sectional curvature).A first attempt to escape the sign limiting of the sectional curvature, as far aswe know, was made by Wang et al. [ ]. However, we did not find in the literature anexample of a Riemannian manifold whose sectional curvature changes sign, endowedwith an (explicit example of a) strictly convex function. In this work, we constructa Riemannian manifold whose sectional curvature assumes any real value greaterthan a negative constant and admits a strictly convex function.Finally, we establish a result relating a geometric property of Riemannian man-ifolds of positive sectional curvature with the set of minimum points of a convexfunction defined on the manifold. We illustrate, with an example, how this resultcan be useful to choose the initial point for an iterative method which seeks theminimum of a convex function.This paper is organized as follows. In Section 2 we introduce some notations,basic definitions and important properties of Riemannian manifolds. In Section 3we prove that the conservativity of the geodesic flow of a Riemannian manifoldwith infinite volume implies that all convex functions on the manifold are constant.In Section 4 we construct a manifold whose sectional curvature assumes any realvalue greater than a negative constant and admits a strictly convex function. InSection 5 we prove that if the set of minimum points of a convex function, definedon a Riemannian manifold of positive sectional curvature, is not empty then we canminimize it using souls of the manifold.
2. Preliminaries
In this section, we present some basics from Riemannian geometry. All themanifolds and vector fields here considered will be assumed to be differentiable(smooth). Let M be a (smooth) manifold, we denote the space of (smooth) vectorfields over M by X ( M ), the tangent bundle of M will be denoted by T M and thering of smooth functions over M by D ( M ). Definition . An r -covariant tensor field ω on a Riemannian manifold M isa D ( M ) -multilinear mapping ω : X ( M ) × · · · × X ( M ) | {z } → D ( M ) .r f actors ONVEXITY AND SOME GEOMETRIC PROPERTIES 3
A couple ( M, h , i ), where M is a manifold and h , i is a smooth metric, (innerproduct on each tangent space varying smoothly on M ), is called a Riemannianmanifold and h , i a Riemannian metric. The metric induces a map f ∈ D ( M ) →∇ f ∈ X ( M ) which associates to each f , its gradient. Let D be the Levi-Civitaconnection associated to ( M, h , i ). The differential of X ∈ X ( M ) is the D ( M )-linear operator A X : X ( M ) → X ( M ), given by A X ( Y ) := D Y X . To each point p ∈ M , we assign the linear map A X ( p ) : T p M → T p M defined by A X ( p ) v = D v X .In particular, if X = ∇ f , then A X ( p ) is the Hessian of f at p and is denoted byHess f .Given a vector field V ∈ X ( M ) on a Riemannian manifold M and an r -covarianttensor field ω , the Lie derivative of ω with respect to V is defined by( L V ω )( X , ..., X r ) = V ( ω ( X , ..., X r )) − r X i =1 ω ( X , ..., [ V, X i ] , ..., X r ) . For instance, if ω = h , i then ( L V h , i )( X, Y ) = h D X V, Y i + h X, D Y V i . We say that V ∈ X ( M ) is a conformal vector field if there exists φ ∈ D ( M ), which is called theconformal factor of V , such that L V h , i = 2 φ h , i . An interesting particular case of a conformal vector field V occurs when D X V = φX for all X ∈ X ( M ), in this case we say that V is closed.Now, let π : T M → M be the projection map. For any normal neighbourhood U of a point p ∈ M there is a canonical map τ : π − ( U ) → T p M defined as follows:for Z = ( q, w ) ∈ π − ( U ), the image τ ( Z ) is obtained by a parallel translationof w along the unique geodesic arc in U joining the point q = π ( Z ) to p . Theconnection map corresponding to D is a map κ : T ( T M ) → T M , inducing for any Z = ( p, z ) ∈ T M a linear map of T Z ( T M ) into T π ( Z ) M , and defined as follows: let A ∈ T Z ( T M ) and ∼ Z : t → ∼ Z ( t ) be a path in T M representing A at t = 0; then κ ( A ) = lim t → τ ( ∼ Z ( t )) − zt . The induced Riemannian metric G in T M (Riemann-Sasaki metric) is deter-mined by the rule G ( A, B ) = h dπ ( A ) , dπ ( B ) i + h κ ( A ) , κ ( B ) i , where A, B ∈ T Z ( T M ) and Z ∈ T M . For more details about Riemann-Sasakimetric see, for instance, Kowalski [ ].To define soul of a Riemannian manifold one needs, firstly, the following con-cepts: a Riemannian submanifold S of M is called totally geodesic if all geodesicsin S are also geodesics in M and it is said to be totally convex if for all points p, q in S , all geodesics joining p to q are contained in S . Definition . Let M be a complete manifold and S ⊂ M a compact totallyconvex, totally geodesic submanifold such that M is diffeomorphic to the normalbundle of S . The submanifold S is called a soul of M . In general the soul is not uniquely determined, but any two souls of M areisometric. Gromoll and Meyer [ ] proved that, when the sectional curvature of M is positive, the soul S of M consists of a single point ( simple point ) and M isdiffeomorphic to an Euclidean space. A point p ∈ M is said to be simple if there J. X. CRUZ NETO , ´ITALO MELO AND PAULO SOUSA are no geodesic loops in M closed at p . In their paper [ ], Gromoll and Meyerproved that the set of simple points in M is open implying that the set of the soulscan not consists of a single point.
3. Conservativity and non-existence of convex functions
Let M be a complete Riemannian manifold and θ = ( p, v ) ∈ T M , and denoteby γ θ ( t ) the unique geodesic with initial conditions γ θ (0) = p and γ ′ θ (0) = v . For agiven t ∈ R , we define a diffeomorphism of the tangent bundle T Mϕ t : T M → T M as follows ϕ t ( θ ) = ( γ θ ( t ) , γ ′ θ ( t )). The family of diffeomorphism ( ϕ t ) is in fact a flow(called geodesic flow ), that is, it satisfies ϕ t + s = ϕ t ◦ ϕ s .Denote by SM the unit tangent bundle of M , that is, the subset of T M givenby those pairs θ = ( p, v ) such that v has norm one. Since geodesics travel withconstant speed, we see that ϕ t leaves SM invariant, that is, given θ ∈ SM thenfor all t ∈ R we have ϕ t ( θ ) ∈ SM . So, ( ϕ t ) preserves the Liouville measure of theunit tangent bundle. The Liouville measure may be described as follows: everyinner product in a finite-dimensional vector space induces a volume element in thatspace, relative to which the cube spanned by any orthonormal basis has volume1. In particular, the Riemannian metrics induces a volume element dv on eachtangent space of M . Integrating this volume element along M , we get a volumemeasure dx on the manifolds itself. The Liouville measure of T M is given, locally,by the product µ = dxdv . If m denotes the Liouville measure restricted to theunit tangent bundle SM , it is known (see, for instance, Paternain [ ]) that m isinvariant under the geodesic flow, i.e., the diffeomorphism ϕ t : SM → SM is ameasure-preserving transformation for all t ∈ R [that is m ( B ) = m ( ϕ − t ( B )) for all B ⊂ SM and t ∈ R ]. Definition . The geodesic flow is conservative with respect to the Liouvillemeasure if, given any measurable set A ⊂ SM , for m -almost all θ ∈ A there existsa sequence ( t n ) n ∈ N in R converging to + ∞ such that ϕ t n ( θ ) ∈ A for all t n . In the class of Riemannian manifolds with finite volume, from Poincar´e recur-rence theorem it follows that the geodesic flow ϕ t : SM → SM is conservative. Thisproperty was used by Yau (see [ ]) to generalize the result of Bishop and O’Neill(see [ ]), Yau proved that there is no non-trivial continuous convex function ona complete Riemannian manifold of finite volume. We emphasize that there areRiemannian manifolds of infinite volume whose geodesic flow is conservative (seeremark below). Remark . A hyperbolic surface is a complete two-dimensional Riemannianmanifold of constant curvature − . Every such surface has the unit disc as uni-versal cover and can be viewed as H/ Γ , where H is the unit disc equipped with thehyperbolic metric and Γ is the covering group of isometries of H . Nicholls [ ] proved that “The geodesic flow on the hyperbolic surface H/ Γ is conservative andergodic if and only if the Poincar´e series of Γ diverges at s = 1 ”. Such surfaces arecalled of divergence type. In [ ] , Hopf proved that geodesic flows on hyperbolic sur-faces of infinite area are either totally dissipative or conservative and ergodic. Thus,surfaces of divergent type with infinite area are examples of Riemannian manifoldswith infinite volume whose geodesic flow is conservative. ONVEXITY AND SOME GEOMETRIC PROPERTIES 5
Lemma . If the geodesic flow ϕ t : SM → SM is conservative with respectto the Liouville measure, then for m -almost all θ ∈ SM there exists a sequence ( t n ) n ∈ N in R converging to + ∞ such that ϕ t n ( θ ) → θ . Proof.
Since SM is a manifold, there is a countable basis { V i } i ∈ N for thetopology of SM such that m ( V i ) < + ∞ , for every i . On the other hand, thegeodesic flow is conservative thus, for every i , there exists a proper subset U i ⊂ V i with m ( U i ) = m ( V i ) satisfying: if θ ∈ V i then there exists a sequence ( t n ) n ∈ N in R converging to + ∞ such that ϕ t n ( θ ) ∈ V i . Now, note that m ( ∼ U ) = 0 where ∼ U = S i ∈ N ( V i \ U i ) and if θ ∈ SM \ ∼ U there exists a sequence ( t n ) n ∈ N in R convergingto + ∞ such that ϕ t n ( θ ) → θ . This concludes the proof. (cid:3) (cid:3) Remark . Cruz Neto et al. [ ] considered the case where M is a com-plete non-compact Riemannian manifold with finite volume, then m ( SM ) < + ∞ and they used the Poincar´e recurrence theorem to get a characterization of the C monotone vector fields. In order to prove our first result we need the following lemma
Lemma . If the geodesic flow ϕ t : SM → SM is conservative with respect tothe Liouville measure, then for m -almost all θ = ( p, v ) ∈ SM there are sequences ( t n ) n ∈ N , ( s n ) n ∈ N in R converging to + ∞ such that ϕ t n ( θ ) → θ and ϕ − s n ( θ ) → θ =( p, − v ) . Proof.
Consider the map T : SM → SM defined by T ( p, v ) = ( p, − v ), it isnot hard to see that T is an isometry with respect to Sasaki metric whose Riemann-ian volume coincides with the Liouville measure. Consider the setΩ = { θ = ( p, v ) ∈ SM : ∃ ( t n ) n ∈ N ⊂ R such that t n → + ∞ and ϕ t n ( θ ) → θ } . Since m ( SM \ Ω) = 0 and T is an isometry, it follows that m ( SM \ T (Ω)) = 0.Therefore, m ( SM \ (Ω ∩ T (Ω))) = m (( SM \ Ω) ∪ ( SM \ T (Ω))) = 0 . Note that if θ = ( p, v ) ∈ Ω ∩ T (Ω) then θ = ( p, v ) , θ = ( p, − v ) ∈ Ω, since ϕ t ( θ ) = ϕ − t ( θ ) and we are done. (cid:3) (cid:3) A function f : M → R on a Riemannian manifold M is convex if its restrictionto every geodesic in M is a convex function along the geodesic, i.e., if for everygeodesic segment γ : [ a, b ] → R and every t ∈ [0 , f ( γ ((1 − t ) a + tb )) ≤ (1 − t ) f ( γ ( a )) + tf ( γ ( b )) . A convex function f is strictly convex if this inequality is strict whenever t ∈ (0 , f is smooth, it is known that f is(strictly) convex provided its hessian is positive (definite) semidefinite, or equiva-lently if ( f ◦ γ ) ′′ ≥ >
0) for every geodesic γ : I ⊂ R → M . Theorem . Let M be a connected complete Riemannian manifold. If thegeodesic flow ϕ t : SM → SM is conservative with respect to the Liouville measure,then all continuous convex functions on M are constant. Proof.
Suppose that f : M → R is a convex function. Note that if γ ( t ) is anygeodesic in M with lim t n → + ∞ f ( γ ( t n )) = lim s n →−∞ f ( γ ( s n )) ∈ R , J. X. CRUZ NETO , ´ITALO MELO AND PAULO SOUSA for sequences ( t n ) , ( s n ) ⊂ R , then f is constant on the geodesic { γ ( t ) : t ∈ R } .As the geodesic flow on M is conservative, then by Lemma 2 it follows thatfor m -almost all point θ = ( p, v ) ∈ SM there are sequences ( t n ) n ∈ N , ( s n ) n ∈ N in R converging to + ∞ such that ϕ t n ( θ ) → θ and ϕ − s n ( θ ) → θ = ( p, − v ).Writing ϕ t ( θ ) = ( γ θ ( t ) , γ ′ θ ( t )), we obtain that for m -almost all θ = ( p, v ) ∈ SM there are sequences ( t n ) , ( s n ) ⊂ R such that t n → + ∞ , s n := − s n → −∞ andlim t n → + ∞ γ θ ( t n ) = p = lim s n →−∞ γ θ ( s n ) . Then, as we noted in the first paragraph of this proof, f must be constant on γ θ ( t ). Now, let θ = ( p, v ) be any point in SM and { θ i } = { ( p i , v i ) } a sequenceconverging to θ where each θ i satisfies the above property. Using the continuity ofthe function f and the fact that the geodesic flow is continuous it follows that f isalso constant on γ θ ( t ). In particular, f is locally constant and by connectedness itfollows that f is constant. (cid:3) (cid:3)
4. Convex functions and the sectional curvature
The known examples of strictly convex functions are associated with the signof the sectional curvature of the Riemannian manifold, for example: • (Theorem 1 (a), Greene and Wu [ ]) If M is a complete non-compactRiemannian manifold of positive sectional curvature, then there exists on M a C ∞ Lipschitz continuous strictly convex function such that, for every λ ∈ R , f − (] − ∞ , λ ]) is a compact subset of M . • (Theorem 4.1, [ ]) Let M be a complete simply connected Riemannianmanifold of non positive sectional curvatures K ≤ S is a closed, totally geodesic submanifold of M , then the C ∞ function f S : M → R defined by f S ( x ) := d ( x, S ) is convex.(b) In (a), if S is a single point p , then f p is strictly convex.However, the sign change of the sectional curvatures does not obstructs theexistence of convex functions. In the following example we present a convex functiondefined on the tangent bundle of a Riemannian manifold. Example . Let ( M, h , i ) be a complete Riemannian manifold and ( T M, G ) its tangent bundle, where G is the Riemann-Sasaki metric induced on T M . A pointin
T M is represented by an ordered pair ( p, v ) , where p ∈ M and v ∈ T p M .Let us consider the kinetic energy E : T M → R defined by E ( p, v ) = h v, v i .It is known in the literature (Theorem 3.6, pp 205, Udriste [ ] ) that E is a C ∞ convex function on ( T M, G ) . Furthermore, the kinetic energy is not strictly convex.In fact, given a geodesic γ ( t ) we get that β ( t ) = ( γ ( t ) , γ ′ ( t )) is a geodesic in T M and E ( β ( t )) is constant implying that the function E : T M → R is not strictlyconvex.Choosing M so that the sectional curvature of T M changes its sign, we havea Riemannian manifold whose sectional curvature changes its sign and admits aconvex function. (cid:3)
In this context, a question naturally arises:
Question.
Does the sign change of curvature imply the non existence of strictlyconvex functions?
ONVEXITY AND SOME GEOMETRIC PROPERTIES 7
In the next theorem we present a geometric condition that ensures the exis-tence of (strictly) convex functions on a particular class of complete non-compactRiemannian manifolds. As a corollary we get the (negative) answer to the raisedquestion. More specifically,
Theorem . Let M be a complete non-compact Riemannian manifold and V ∈ X ( M ) a closed conformal vector field with conformal factor φ . If h∇ φ, V i ≥ ,then the energy function f = h V, V i is convex. In addition, if the vector field V and the function φ never vanishes then f is a strictly convex function. Proof.
For every vector fields
X, Y ∈ X ( M ) we have X ( f ) = h D X V, V i = h φX, V i , then ∇ f = φV . Now, note thatHess f ( X, Y ) = h D X ∇ f, Y i = h D X ( φV ) , Y i = h X, ∇ φ ih V, Y i + h φD X V, Y i = h X, ∇ φ ih V, Y i + φ h X, Y i . Since both Hess f and the metric are symmetric tensors, we deduce h X, ∇ φ ih V, Y i = h Y, ∇ φ ih V, X i for all X, Y ∈ X ( M ). So | V | · ∇ φ = h V, ∇ φ i · V. Therefore, | V | · Hess f ( X, X ) = h V, ∇ φ ih V, X i + φ | V | | X | ≥ φ | V | | X | . Since the set of the points of M where V vanishes is a discrete set (see, forinstance, Montiel [ ]), we conclude the proof of the theorem. (cid:3) (cid:3) Now, let us apply Theorem 2 to construct examples of strictly convex functionson Riemannian manifolds whose sectional curvature assumes any real value greaterthan a negative constant. We first describe some preliminaries which will be usefulto understand the construction.Let ( B, h , i B ) and ( P, h , i P ) be Riemannian manifolds and g > B . Set M = B × P (with the structure of product manifold),and let π B : M → B and π P : M → P denote the canonical projections; we equip M with the warped metric h , i , given by h X, Y i = h dπ B ( X ) , dπ B ( Y ) i B + ( g ◦ π B ) h dπ P ( X ) , dπ P ( Y ) i P and denote the resulting Riemannian space by M = B × g P . We remark that thewarped metric will be complete, for any g as above if, and only if B and P arecomplete.In this case, B is called the base of M and P , the fiber. It is easy to see thatthe fibers { b } × P = π − B ( b ) and the leaves B × { p } = π − P ( p ) are Riemanniansubmanifolds. Tangent vectors to the leaves are called horizontal and tangentvectors to the fibers, vertical . J. X. CRUZ NETO , ´ITALO MELO AND PAULO SOUSA Proposition . [Proposition 42 (5), O’Neill [ ]] Let M = B × g P be a warpedproduct with Riemannian curvature tensor R and let R P be the lift to M of theRiemannian curvature tensor of P . If U, V, W ∈ T M are vertical, then R ( V, W ) U = R P ( V, W ) U − |∇ g | g ( h V, U i W − h W, U i V ) . If U, V ∈ T M are orthonormal, K and K P denote the sectional curvature of M and P we get K ( U, V ) = 1 g (cid:0) K P ( U, V ) − |∇ g | (cid:1) . An interesting class of spaces furnished with closed conformal vector fields isgiven by the following subclass of warped product spaces: when B = I ⊂ R is anopen interval.If ∂ t = ∇ π I is the standard unit vector field on I , then V = ( g ◦ π I ) ∂ t is aclosed conformal nowhere vanishing vector field on this manifold, with conformalfactor φ = g ′ ◦ π I . Therefore, ∇ φ = ( g ′′ ◦ π I ) ∂ t . In this case, the hypothesis of theTheorem 2 is equivalent to0 ≤ h V, ∇ φ i = h ( g ◦ π I ) ∂ t , ( g ′′ ◦ π I ) ∂ t i = ( g ◦ π I )( g ′′ ◦ π I ) . Thus, making use of Theorem 2 we conclude that the function f : M = I × g P → R defined by f = h V, V i is convex since that g ′′ ≥ P = { ( x, y, z ) : z = x + y } , B = R and g ( t ) = e t . From Proposition 1 the manifold M = R × e t P has vertical sectional curvature K ( t, x, y ) = 1 e t (cid:18) x + 4 y ) − e t (cid:19) .M is a Riemannian manifold whose sectional curvature assumes any real valuegreater than − φ ( t, x, y ) = | V ( t, x, y ) | = e t = 0 for all t ∈ R .We summarize the above construction with the following corollary. Corollary . There is Riemannian manifold whose sectional curvature as-sumes any real value greater than a negative constant and admits a strictly convexfunction.
5. Soul of a manifold and minimization of convex functions
In order to prove our third result we need the following lemma.
Lemma . [Theorem 1 (a),[ ]] If M is a complete non-compact Riemannianmanifold of everywhere positive sectional curvature, then there exists on M a C ∞ Lipschitz continuous strictly convex function such that, for every λ ∈ R , f − (] −∞ , λ ]) is a compact subset of M . Theorem . Let M be a complete non-compact Riemannian manifold of pos-itive sectional curvature and let u : M → R be a convex function admitting a point p ∈ M such that u ( p ) = inf M u > −∞ . Then, there exists a sequence ( x k ) suchthat x k is a soul of M , x k → p and u ( p ) = inf M u . ONVEXITY AND SOME GEOMETRIC PROPERTIES 9
Proof.
Let f be the function whose existence is assured by the previouslemma. Since f − (] − ∞ , λ ]) is a compact subset of M , for every λ ∈ R , we havethat inf M f > −∞ . Consider the function g : M → R defined by g := f − inf M f ,then g ≥
0. We can assume, without loss of generality, that u ≥
0. For k ∈ N fixed,consider the function h k := k · u + g and note that for r ∈ R we have x ∈ h − k (] − ∞ , r ]) ⇔ h k ( x ) ≤ r ⇔ ku ( x ) + g ( x ) ≤ r. Then x ∈ h − k (] − ∞ , r ]) ⇒ g ( x ) ≤ r ⇒ f ( x ) ≤ r + inf M f . Hence we concludethat h − k (] −∞ , r ]) ⊂ f − (] −∞ , r +inf M f ]), since f − (] −∞ , r +inf M f ]) is compactit follows that h − k (] − ∞ , r ]) is also. On the other hand, since h k is strictly convexthere exists a unique point x k such that h k ( x k ) = inf M h k . Now, fixed p such that u ( p ) = inf M u = 0 we get(1) k · u ( x k ) + g ( x k ) = h k ( x k ) ≤ h k ( p ) = g ( p ) . From the above inequality it follows that 0 ≤ g ( x k ) ≤ g ( p ), so we get x k ∈ g − (] − ∞ , g ( p )]) = f − (] − ∞ , g ( p ) + inf M f ]). As this is a compact set, we have x k → p ∈ M or we can consider a subsequence of ( x k ) if necessary. Therefore, from(1) we have 0 ≤ u ( x k ) ≤ g ( p ) k and thus we obtain u ( p ) = inf M u = 0. Since h k is a strictly convex function wehave that the set { x k } is totally convex, from [Theorem 2,[ ]] it follows that every x k is a soul of M . (cid:3) (cid:3) In optimization it is important to establish the convergence of numerical meth-ods to find minimum points for convex functions and it is crucial the initial pointin the iterative process. The next result illustrates the region where we shouldstart the iterative process when the Riemannian manifold to be considered is aparaboloid.
Corollary . Let u be a convex function defined on the paraboloid P = { ( x, y, z ) : z = x + y } such that the minimum set is non empty, then thereexists a minimizer p = ( x , y , z ) for u such that z ≤ β , where β = q (1 + µ ) and µ − arctan µ = π . Proof.
Let h : R → R be the projection on the third coordinate, i.e., h ( x, y, z ) = z . We recall that a point p ∈ P is simple if there are no geodesicloops in M closed at p . Following the terminology of Ling and Recht [ ], p isa simple point (a Soul) is equivalent to saying that p is not vertex for any loop.Moreover, in [ ] the authors showed that p is not vertex for any loop if, only if, h ( p ) < β .From Theorem 3, there is a souls-sequence ( p k ) of P such that p k → p where u ( p ) = min P u . As a consequence of the result proved by Ling and Recht, we get h ( p k ) < β . Since h is a continuous function, we conclude that h ( p ) ≤ β . (cid:3) (cid:3)
6. Conclusions
In Theorem 1 we have proved that the conservativity of the geodesic flow on aRiemannian manifold M with infinite volume implies that all convex functions on M are constant, this fact generalizes in a certain sense the result proved by Yau [ ].In the Theorem 2 we presented a geometric condition that ensures the existence , ´ITALO MELO AND PAULO SOUSA of (strictly) convex functions on a particular class of complete non-compact Rie-mannian manifolds. Finally, in the Theorem 3 we have related the geometry of aRiemannian manifold with the set of minimum points of a convex function definedon the manifold. References [1] Bishop, R.L., O’Neill, B.: Manifolds of negative curvature, Transactions AMS (1969),1–49.[2] Yau, S.T.: Non-existence of continuous convex functions on certain Riemannian manifolds,Math. Ann., (1974), 269–270.[3] Shiohama, K.: Convex Sets and Convex Functions on Complete Manifolds, Proceedings ofthe International Congress of Mathematicians, Helsinki (1978).[4] Cruz Neto, J.X., Lima, L.L., Oliveira, P.R.: Geodesic algorithms in Riemannian geometry.Balkan J. Geom. Appl. (2) (1998), 89–100.[5] Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim.Theory Appl. (1) (1998), 93–104.[6] Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimiza-tion (2) (2002), 257–270.[7] Wang, X.M., Li, C., Yao, J.C.: Subgradient Projection Algorithms for Convex Feasibilityon Riemannian Manifolds with Lower Bounded Curvatures. J. Optim. Theory Appl. (1)(2015), 202–217.[8] Kowalski, O.: Curvature of the Induced Riemannian Metric on the Tangent Bundle of aRiemannian Manifold, Journal fur die reine und angewandte, (1971), 124–129.[9] Gromoll, D., Meyer, W.: On complete open manifolds of positive curvature, Ann. of Math., (1969), 75–90.[10] Paternain, G.P.: Geodesic Flows, Progress in Mathematics, Birkh¨auser (1999).[11] Nicholls, P.: Transitivity properties of Fuchsian groups, Canad. J. Math., (1976), 805–814.[12] Hopf, E.: Ergodentheorie, Chelsea, 1948.[13] Cruz Neto, J.X., Melo, I.D., Sousa, P.A.: Non-existence of Strictly Monotonous Vector Fieldson Certain Riemannian Manifolds, Acta Math. Hungar., (2015), 240–246.[14] Greene, R.E., Wu, H.: C ∞ convex functions and manifolds of positive curvature, Acta Math-ematica n.1 (1976), 209–245.[15] Udriste, C.: Convex functions and optimization methods on Riemannian manifolds, Mathe-matics and its applications v 297, 1994.[16] Montiel, S.: Stable constant mean curvature hypersurfaces in some Riemannian manifolds,Commentarii Mathematici Helvetici, (1998), 584–602.[17] O’Neill, B.: Semi-Riemannian Geometry, Academic Press, 2010.[18] Ling, D., Recht, L.: A theorem concerning the geodesics on a paraboloid of revolution, Bull.Amer. Math. Soc.,47