Geodesics without conjugate points and curvatures at infinity

Geodesics without Conjugate Points and Curvatures at Infinity

SÉRGIO MENDONÇA and DETANG ZHOU

Universidade Federal Fluminense - UFF, Instituto de Matemática,

Campus do Valonginho, rua Mário Santos Braga s/n, 7º andar - 24020-140 Niterói, RJ

Manuscript received on July 16, 1999; accepted for publication on February 8, 2000;

presented by MANFREDO DO CARMO

ABSTRACT

We study the asymptotic behavior of curvature and prove that the integral of curvature along a geodesic without conjugate points is nonpositive and some generalizations of Myers theorem and Cohn-Vossen's theorem. Some applications are also given.

Key words: Riemannian manifold, geodesic, conjugate point.

1. MAIN RESULTS

Let Mⁿ e an n-dimensional Riemannian manifold and let d(x,y) be the distance induced by the metric. (Ambrose 1957) showed that if the integral of the Ricci curvature along a geodesic is infinite then there is a such that is conjugate to . We extended this result in two directions: first we obtain

THEOREM A. Let be a geodesic without conjugate points (particularly if is a line). Then for any unit vector field which is parallel along it holds that

where is the sectional curvature of the plane spanned by and . Moreover, if then . If then for all and any orthogonal to , it holds that .

Note that is not supposed to be complete in Theorem A, and no hypothesis on the curvature is assumed. It should be remarked that (Liang & Zhan 1996) proved that if is a geodesic without conjugate points and if , then .

Recall that the minimal radial curvature if for any minimal normal geodesic joining and and any unit vector orthogonal to . We say that the radial Ricci curvature if for any normal (not necessarily minimizing) geodesic joining and it holds that . As a corollary of Theorem A we have

COROLLARY 1. Assume that M is a complete manifold without conjugate points satisfying Ric_o³0, for a certain base point . Then M is flat, that is, M is isometric to .

It should be remarked that when M is compact and the condition Ric_o³0 is replaced by the scalar curvature S ³0, the same conclusion of Corollary 1 follows from (Green 1958).

In order to state our second result we need some definitions. In the following M = Mⁿ always denotes a complete and connected -dimensional Riemannian manifold. is always a connected manifold without boundary which is isometrically immersed and whose image is closed in M. This assumption is weaker than that the immersion being proper, since if is any compact manifold and has no conjugate points, then the exponential map is an immersion whose image is closed, but the inverse image of of course is not compact. Unless otherwise stated all geodesics are supposed to be normalized. One of the subjects treated here is to study the manifolds with minimal -radial (Ricci) curvature bounded from below (or from above). Even if we strengthen our curvature conditions with the corresponding ones on the sectional (or Ricci) curvature some of our results are new.

The notion of minimal radial curvature was first introduced by (Klingenberg 1963) and was studied by many authors. It is natural to extend such definition for submanifolds when we study existence of minimal submanifolds. The notion of minimal -radial curvature appears - even without an explicit definition - for example in (Eschenburg 1987) and in (Heintze & Karcher 1978). Given we say that a minimal geodesic is a minimal connection between and if and the distance . Given linearly independent tangent vectors we denote by the sectional curvature associated with the plane generated by and .

0.1.DEFINITION. Given , we say that the minimal -radial curvature if for any minimal connection between and , and any orthogonal to the tangent vector at it holds that . We say similarly that the minimal -radial Ricci curvature if any orthonormal frame which is orthogonal to at satisfies . Finally we say that the parallel minimal -radial Ricci curvature if , where are obtained by the parallel transport along of an orthonormal basis of .

Note that if the dimension of is then is equivalent to . We can give some examples of radial curvature bounded from below. One of the most well-known results relating the curvature and topology of a complete Riemannian manifold is the classical Theorem of (Myers 1941) which states that if the Ricci curvature with respect to unitary vectors in has a positive lower bound then is compact. For the distance function set . The following result shows that, if , then the existence of a minimal submanifold implies that radial curvatures tend to be nonpositive in some integral sense. Precisely we have:

THEOREM B. Let be minimal and have dimension . Assume that satisfies for any , where is supposed to be a continuous function. Then the condition implies that Furthermore, if then we have , for all .

We remark that can be finite even if is noncompact. For example, let be a line in a cylinder . When Theorem B implies that, if as above is nonnegative, then . This fact however does not imply that . It means only that for any it holds that (see for example the case in that is a meridian of a paraboloid).

If a ray satisfies and we will say that is an - ray. By the same proof as in Theorem A we obtain the following result.

COROLLARY 2. Assume that is totally geodesic (respectively, minimal), has dimension , and that is an -ray. Let be a parallel field along with . Then we have (respectively, ). If this integral limit vanishes then (respectively, ).

The Theorems of (Cohn-Vossen 1935) and (Huber 1957) assert that if and the negative part of its Gaussian curvature is integrable, then , where is the volume element of , and is the Euler characteristic of . This implies that for this type of manifolds there does not exist a sequence of points with in the ball with center , radius , and volume , for fixed positive numbers and . It should be noted that this is false without the assumption on the integrability of the curvature. This can be seen if we consider with the periodic metric induced by the universal covering of a nonflat metric on torus. It has been asked by many mathematicians about the extension of theorems of Cohn-Vossen and Huber to the higher dimensions (see for example Yau 1991).

As pointed out before we first need some suitable integrability conditions about curvatures. It seems for us that the asymptotically nonnegative condition studied extensively by (Abresch 1985) is one reasonable choice. In dimension let be the infimum of the sectional curvatures at the point . It would be interesting to obtain an integral inequality for the function . However we have only obtained the nonexistence of a sequence as above in this case. We recall (see Abresch 1985) that the curvature of a complete manifold is asymptotically nonnegative if there exists a nonincreasing function such that , and , for a fixed point . Abresch obtained (Abresch 1985) a version of Toponogov Theorem for this class of manifolds, and Kasue constructed in (Kasue 1988) a compactification of such manifolds . Our third theorem is the following.

THEOREM C. Let be a complete manifold with asymptotically nonnegative curvature. Take a sequence . Suppose that in , with , where is a fixed number. Then .

In particular it is not possible that in , if the volume of is a constant . In fact, let be the minimum of in . Because of the Bishop-Gromov comparison volume Theorem we would have in that case , where is the radius of a ball of volume in the hyperbolic space of constant curvature . So we would not have , and this contradicts Theorem C. We can give an example to show that the condition is essential in Theorem C, even if .

0.2. DEFINITION. An embedded submanifold is said to be a polar submanifold if the normal exponential map is a diffeomorphism.

0.3. DEFINITION. is called a geometric soul if the distance function is convex.

Clearly any geometric soul is totally convex (any geodesic joining two points of is contained in ). In (Cheeger & Gromoll 1972) it is proved that any totally convex set is of the form , where is a embedded submanifold and is a boundary of class. Since our has no boundary we conclude that any geometric soul is of class and totally geodesic. For the case of nonnegative -radial curvature we have the following result.

THEOREM D. Assume that is a polar submanifold of and that . Given and a unitary vector it holds that the Hessian

hence the function is convex and is a geometric soul. In fact even if is not polar the conclusion is valid in all point outside the cut locus of .

Using Theorem D we can prove:

COROLLARY 3. Let satisfy (or and be totally geodesic). Then the distance function is convex if and only if is polar. In particular, under these conditions the distance function from a point is convex if and only if is a pole.

The following result is a generalization of the famous result of (Frankel 1966) about the fundamental group of positively curved manifolds. We remark that in our case the fundamental group can be infinite, since the curvature of can be negative. We say that the Ricci radial curvature when for any geodesic (not necessarily minimizing) with it holds that (g', g') ³ c.

THEOREM E. Assume that is a compact minimal hypersurface in and that (or instead and ). Then the natural homomorphism of fundamental groups is onto.

2. OUTLINE OF THE PROOFS

Let be a Riemannian manifold of dimension . Let be a geodesic without conjugate points. Consider the normal bundle associated with the isometric immersion . Take a closed neighborhood of in such that the normal exponential map is a local diffeomorphism. We consider in the Riemannian metric induced by . Let be the trivial curve given by . We define the distance functions as

where dist is the intrinsic distance in . Since there are no conjugate points in , for sufficiently small we have that is smooth at and , for any . Also it is easy to see that , for all . For any unit vector field which is parallel along , consider the Hessian . Given some curve with and , for small we have geodesics joining and with . So we have . Thus a direct calculation shows that

(1.1)

LEMMA 1.1. With the notations above exists for every . The derivative exists and .

To prove Theorem A we need the following lemmas, which are stated here without proofs. For a positive constant , consider the following inequality of Ricatti type:

(1.2)

LEMMA 1.2. If for a constant and is a solution of (1.2) on , then .

LEMMA 1.3. If is a solution of (1.2) for , then

(1.3)

We have if and only if and .

Now we are in the position to prove our theorem.

PROOF OF THEOREM A. It follows from (1.1) and Lemma 1.1 that

(1.4)

Therefore it follows immediately from Lemma 1.3 that and that if and only if .

Take an arbitrary with which is orthogonal to . Consider an orthonormal basis , and the parallel transport of along . For each consider the sectional curvature . Then we have the existence of functions such that

(1.5)

Set . Using the inequality

we get

(1.6)

Therefore by Lemma 1.3 we have and that if and only if and .

From (1.5) we have

(1.7)

Since and , we have from (1.7) that for all . Using (1.5) again we conclude that , hence .

Theorem A is proved.

• 

To prove Theorem B we use the same idea and the following lemma.

LEMMA 1.4. The following boundary value problem

(1.8)

has no positive solution on , if and , for any .

SKETCH OF THE PROOF OF THEOREM C. Take a sequence as above. Assume by contradiction that , for some constant . Set . Let be a constant such that

Let , where , where is the boundary of the ball . We can prove that there exists a point , such that . Let be a minimal geodesic joining and . Consider a minimal connection between and , for . Set and . It is not difficult to obtain that . After a series of lemmas we show that

For sufficiently large , we have , and . So we obtain , because of the monotonicity of the function . We obtain also that . Thus we conclude that

Replacing by its value we conclude that

and proves Theorem C.

• 

SKETCH OF THE PROOF OF THEOREM D. We can reduce again the estimate of Hessian of the distance function to estimate of solutions of Ricatti equations.

, there is a representative loop

ACKNOWLEDGMENTS

The authors would like to thank professor Manfredo do Carmo for his encouragement and support, professor Fuquan Fang for useful remarks and professor Francesco Mercuri for pointing to us a mistake in a previous version of this paper. The second named author would like to thank IMPA and UFF for the hospitality, while the work was done. Finally we acknowledge the financial support provided by CNPq.

Correspondence to: Detang Zhou

E-mail: [e-mail]

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Publication Dates

• Publication in this collection
  07 Aug 2000
• Date of issue
  June 2000

History

• Accepted
  08 Feb 2000
• Received
  16 July 1999