Myers’ type theorem with the Bakry–Émery Ricci tensor

In this paper, we prove a new Myers’ type diameter estimate on a complete connected Reimannian manifold which admits a bounded vector field such that the Bakry–Émery Ricci tensor has a positive lower bound. The result is sharper than previous Myers’ type results. The proof uses the generalized mean curvature comparison applied to the excess function instead of the classical second variation of geodesics.     

Some Results on Contact Metric Manifolds

If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.     

Complete gradient shrinking Ricci solitons have finite topological type

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:(i) the Ricci curvature is bounded from above;(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. To cite this article: F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Non-trivial m-quasi-Einstein metrics on quadratic Lie groups

We call a metric m-quasi-Einstein if \({Ric_X^m}\) (a modification of the m-Bakry–Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contain Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field X such that the left-invariant Riemannian metric on a quadratic Lie group is m-quasi-Einstein is a Killing vector field. Then we construct infinitely many non-trivial m-quasi-Einstein metrics on solvable quadratic Lie groups G(n) for m finite.     

Invariant metrics of positive Ricci curvature on principal bundles

Let be a compact connected Riemannian manifold with a metric of positive Ricci curvature. Let be a principal bundle over with compact connected structure group . If the fundamental group of is finite, we show that admits a invariant metric with positive Ricci curvature so that is a Riemannian submersion. Received 14 January 1997     

Bundle stabilisation and positive Ricci curvature

If E is the total space of a vector bundle over a compact Ricci non-negative manifold, it is known that E×Rp admits a complete metric of positive Ricci curvature for all sufficiently large p. In this paper we establish a small, explicit lower bound for the dimension p.     

A Myers-type theorem and compact Ricci solitons

Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci solitons.

     

Prescribing Curvature Problems on the Bakry-Emery Ricci Tensor of a Compact Manifold with Boundary

t The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Amp~re type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.     

Infinitesimal harmonic transformations and ricci solitons on complete Riemannian manifolds

Ricci solitons were introduced by R. Hamilton as natural generalizations of Einstein metrics. A Ricci soliton on a smooth manifold M is a triple (g₀,ξ, λ), where g₀ is a complete Riemannian metric, ξ a vector field, and λ a constant such that the Ricci tensor Ric₀ of the metric g₀ satisfies the equation ò2 Ric₀ = L_ξg₀ + 2λ_go. The following statement is one of the main results of the paper. Let (g₀,ξ, λ) be a Ricci soliton such that M,g₀ is a complete noncompact oriented Riemannian manifold, $ \int\limits_M {\left\| \xi \right\|dv < \infty } $ \int\limits_M {\left\| \xi \right\|dv < \infty } , and the scalar curvature s₀ of g₀ has a constant sign on M, then (M, g₀) is an Einstein manifold     

Killing Vector Fields and Twistor Forms on Generalized Sasakian Space Forms

We study generalized Sasakian space form M(f ₁, f ₂, f ₃) when (i) the Reeb vector field of the almost contact metric structure is Killing, (ii) the Ricci tensor satisfies Einstein-like conditions and (iii) the fundamental 2-form of the almost contact metric structure is a twistor form.     

A Class of Conformally flat Contact Metric 3-Manifolds

It is shown that a conformally flat contact metric 3-manifold with Ricci curvature vanishing along the characteristic vector field, has non-positive scalar curvature. Such a manifold is flat if (i) it is compact, or (ii) the scalar curvature is constant, or (iii) the norm of the Ricci tensor is constant.     

Hamilton-Souplet-Zhang’s gradient estimates for two weighted nonlinear parabolic equations

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-E′mery Ricci tensor bounded below: One is u_t=Δ_fu + aulogu+bu with a, b two real constants, and another is u_t = Δ_fu+λu~α with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.     

Ricci curvature and betti numbers

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower bound. We also give a uniform estimate on the generators of the fundamental group and prove a fibration theorem in this setting.     

第四类Caftan-Hartogs域上Bergman度量与Einstein-Kahler度量等价

In this paper,we discuss the invariaut complete metric on the Cartan-Hartogs domain of the fourth type.Firstly,we find a new invariant complete metric,and prove the equivalence between Bergman metric and the new metric;Secondly,the Ricci curvature of the new metric has the super bound and lower bound;Thirdly,we prove that the holomorphic sectional curvature of the new metric has the negative supper bound;Finally,we obtain the equivalence between Bergman metric and Einstein-Kahler metric on the Cartan-Hartogs domain of the fourth type.     

Quantitative Isoperimetry à la Levy-Gromov

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the manifold and of the corresponding sphere is likewise bounded. These results are actually obtained in the more general context of (possibly nonsmooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method. © 2018 Wiley Periodicals, Inc.     

On the fundamental group of manifolds with almost nonnegative Ricci curvature

Gromov conjectured that the fundamental group of a manifold with almost nonnegative Ricci curvature is almost nilpotent. This conjecture is proved under the additional assumption on the conjugate radius. We show that there exists a nilpotent subgroup of finite index depending on a lower bound of the conjugate radius.

     

On the lower bound of energy functionalE 1 (I)—A stability theorem on the Kähler Ricci flow

In the present article, we prove a stability theorem for the Kaehler Ricci flow near the infimum of the functional E₁ under the assumption that the initial metric has Ricci >−1 and ⋎Riem÷ bounded. At present stage, our main theorem still need a topological assumption (1.2) which we hope to be removed in subsequent articles. The underlying moral is: If a Kaehler metric is sufficiently closed to a Kaehler Einstein metric, then the Kaehler Ricci flow converges to it. The present work should be viewed as a first step in a more ambitious program of deriving the existence of Kaehler Einstein metrics with an arbitrary energy level, provided that this energy functional has a uniform lower bound in this kaehler class.     

Classification of locally rotationally symmetric Bianchi-I space–times using conformal Ricci collineations

We present a complete classification of locally rotationally symmetric (LRS) Bianchi-I space–times in accordance with their conformal Ricci collineations (CRCs). In the case where the Ricci tensor is nondegenerate, we find a general form of the vector field generating CRCs subject to some integrability conditions. Solving the integrability conditions in different cases, we find that the LRS Bianchi-I space–times admit 7-, 10-, 11-, or 15-dimensional Lie algebras of CRCs in the case where the Ricci tensor is nondegenerate. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We give some examples of LRS Bianchi-I space–times that admit nontrivial CRCs and are models of a perfect fluid.     

Short-Time Existence for Some Higher-Order Geometric Flows

We establish short-time existence and regularity for higher-order flows generated by a class of polynomial natural tensors that, after an adjustment by the Lie derivative of the metric with respect to a suitable vector field, have strongly parabolic linearizations. We apply this theorem to flows by powers of the Laplacian of the Ricci tensor, and to flows generated by the ambient obstruction tensor. As a special case, we prove short-time existence for a type of Bach flow.     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.