Manifolds with Bakry-Emery Ricci Curvature Bounded Below

In this paper we show that, under some conditions, if M is a manifold with BakryÉmery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-Émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.


Introduction
Let ( ) , M g be a complete Riemannian manifold and : f M R → a smooth function.
A Bakry-Émery Ricci curvature is defined by When f is a constant function, the Bakry-Émery Ricci tensor becomes the Ricci tensor so it is natural to investigate which geometric and topological results for the Ricci tensor extend to the Bakry-Émery Ricci tensor.
As an extension of Ricci curvature, many classical results in Riemannian geometry asserted in terms of Ricci curvature have been extended to the analogous ones on Bakry-Émery Ricci curvature condition.
In [1] G. Wei and W. Wylie proved some comparison theorems for smooth metric measure spaces with Bakry-Émery Ricci tensor bounded below.In this paper we establish a Myers type theorem for manifolds bounded below by a negative constant.Therefore we prove that is a generalization of the theorem of M. Limoncu in [2] or H. Tadano in [3].
In the second part of this paper we establish a condition on noncompact manifold with nonnegative Bakry-Émery Ricci curvature to be diffeomorphic to the euclidean space n  .

Mains Results
The following theorem is a similar theorem proved in [4] and [5] and is a generalization of Myers theorem.Theorem 2.1.Let ( ) , ,e f g M g dvol − be a metric space such that Suppose that M contains a ball ( ) 0 , B x r of center 0 x and radius r such that the mean curvature ( ) m r of the geodesic sphere ( ) 0 , S x r with respect the inward pointing normal vector verifies ( ) ( ) If there exists a constant c ≥ 0 such that f c ≤ then M is compact and It is well known that there exist noncompact manifolds with nonnegative Ricci curvature which are not finite topological type.Recall that a manifold M is said to have finite topological type if there is a compact domain Ω whose boundary ∂Ω is a topo- logical manifold such that \ M Ω is homeomorphic to [ ) 0, ∂Ω × +∞ .An important result about topological finiteness of a complete Riemannian manifold M is due to Abresch and Gromoll (See [6]).
Let f be a potential function on M satisfying ( ) ( ) for some nonnegative constant c and a fixed point p.
Set ( ) ( ) ( ) In this paper we show a topological rigidity theorem for noncompact manifolds with nonnegative Bakry-Émery Ricci curvature as follow: Theorem 2.2.Let ( ) , ,e f g M g dvol − be a metric space such that ( ) then M is diffeomorphic to n  .

Proofs
Proof of theorem 2.1.The techniques used in the proof of this theorem are based on 1 m S − be the unit sphere in m  and take a real r and with initial values ( ) 0 r a φ = and we define a Riemannian metric tensor by where Thus the Riemannian incomplete manifold with mean curvature vector with outward pointing vector i.e. with pointing positive s Now let prove, under the hypotheses of theorem2.Y can be extended to a Jacobi field along γ , null at p.
In the geodesic polar coordinates the volume element can be written as: where is the volume form on the unit sphere , d , To prove the theorem 2.1 we use the following theorem proved by G. Wei and W.
Theorem 3.1.(Mean Curvature Comparison).Let p be a point in M. Assume along that minimal geodesic segment from p. Equality holds if and only if the radial sectional curvatures are equal to H and ( ) ( ) ≤ along a minimal geodesic segment from p and along that minimal geodesic segment from p.
3) If f c ≤ along a minimal geodesic segment from p and In particular when 0 H = we have ( ) where 4 n c H m + is the mean curvature of the geodesic sphere in In fact in [1] G. Wei and W. Wylie stated that, if where ( )

y t Hy t ′′ + =
From theorem 3.1 above and Equations (( 8) and ( 9)) for all s r ≥ , we have: , If ( ) ( ) Hence there exists 0 R R ≤ so that ( ) 0 , 0 A R θ = which means that there exists 0 i so that the ( ) is a conjugate point of the center p of the sphere ( ) , S p r .Hence γ ceases to be minimal, that is ( ) ( ) ( ) In [2] M. Limoncu generalized a classical Myers theorem by using the Bakry-Émery Ricci curvature tensor on complete and connected Riemannian manifolds ( ) This theorem can be viewed as a corollary of theorem 2.1.Corollary 3.2.Let (M, g) be a complete and connected Riemannian manifold dimension n.If there exists a smooth function : f M →  satisfying the inequalities and f c ≤ then M is compact.

Proof of Corollary
To prove this corollary it suffices to show that there exist a positive real  with k <  and a geodesic sphere ( ) , S p r which mean curvature verifies ( ) ( ) where ( ) ( ) which allows that By Compactness of ( ) , S p r , there exists a positive constant ′  so that, for any geodesic γ emanating from p we have ( ) ( ) ≥ − for all t r ≥ then p admits a conjugate point along γ .Hence, if M is noncompact, for all p M ∈ , there exists a geodesic γ issuing from p so that for any two positive real k and r there exists t r and the conclusion follows.
Corollary 3.4.(Ambrose) Let ( ) , M g be a complete and connected Riemannian manifold of dimension n.
Suppose there exists a function f on M so that 0 f Ric ≥ .If there exists a point p in M so that, for any geodesic γ emanating from p, parametrized by it's arc-length we have then M is compact.

Proof
If M is noncompact, from corollary 3.3, there exists 0 0 r > so that ( ) .
By the second variation formula we have: ( . Hence ( ) (9) and the above relation, we For all positive reals r and s, integrating this relation we have: . .
and integrating from 0 to R′ with respect to s we obtain the conclusion.
Set ( ) ( ) ( ) .   ( ) We say that M is of large weighted volume growth if By lemma 3 in [7] we have:

R cut R h h h r r cut
hence, there exists a ray γ issuing from p verifying ( ) Let q be a point on γ so that ( ) ( ) The inequalities (43) and (47) show that x is not a critical point of p d .Hence, by isotopy lemma M is diffeomorphic to n  .
Hessf denotes the Hessian of f.The function f is called the potential function.For simplicity, denote g Ric by Ric .The Bakry-Émery tensor occurs in many different subjects, such as diffusion processes and Ricci flow.

Y 1 γ . The geodesic 1 γ
is a ( ) , S p r -Jacobi field along can be extend to a minimal geodesic γ starting at p: ( ) = (see[4], Proposition 3) and i Y is a ( ) , S p r -Jacobi field along 1 γ if and only if i

M
+ the simply con- nected model space of dimension 4 n c + with constant curvature H and H m is the mean curvature of the model space of dimension n.
volume of the geodesic ball of center p and radius s in M and ( ) m H vol s the volume of geodesic ball of radius s in the model space m H M with constant curvature H and dimension m.In Differential Geometry, the volume comparison theory plays an important rule.Many important results in this topic can not be obtained without volume comparison results as topological rigidity results.For complete smooth metric measure space with 0 f Ric ≥ the following lemma improved the volume comparison theorem proved by G. Wei and W. Wylie In [orthonormal vector fields along γ orthonormal to γ .

For
by part (1) of the lemma 3.6 we have: the conclusion follows from theorem 2.1.
be the set of the unit initial tangent vectors to the geodesics starting from p which are minimized at least to t and [10] the inequality (28) and using the arguments of the proof of the Proposition 2.3 in[6], we deduce the following excess estimate for complete smooth metric measure By the same arguments as in[10]and using h ∆ instead of ∆ , one can prove the above lemma.To prove the theorem 2.2, it suffices to show that M contains no critical point of p d other than p.For this, x be a point in M and x p [8] the part (2) can be proved as the lemma 3.10 in[8].p d .Then for any ray γ issuing from p, we have ( ) ( ) Recall that a point x is a critical point of p d if for any vector x u T M