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.
Riemannian manifold; geodesic; conjugate point
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 Rico³0, 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 Rico³0 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: zhou@impa.br
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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