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We give a sufficient condition for the essential self-adjointness of a perturbed biharmonic-type operator acting on sections of a Hermitian vector bundle on a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a (possibly unbounded) non-positive function depending on the distance from a reference point. We also establish the separation property in the case when the corresponding operator acts on functions.

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections and classify them when the covariant derivatives associated with both connections, either in the direction of the structure vector field or any direction of the maximal holomorphic distribution, coincide when applying to the Riemannian curvature tensor of the real hypersurface.

We study minimal graphic functions on complete Riemannian manifolds ∑ with nonnegative Ricci curvature, euclidean volume growth, and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville-type theorem with such growth via splitting for tangent cones of ∑ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville-type theorem under a certain nonradial Ricci curvature decay condition on ∑. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section. © 2014 Wiley Periodicals, Inc.

The property of being a D'Atri space (i.e., a Riemannian manifold with volume-preserving geodesic symmetries) is equivalent, in the real analytic case, to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition L3 is said to be an L3-space. This definition extends easily to the affine case. Here we investigate the torsion-free affine manifolds and their Riemann extensions as concerns heredity of the condition L3. We also incorporate a short survey of the previous results in this direction, including also the topic of D'Atri spaces.

In the first part of this paper, we prove the sharp global Li-Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman-type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu–Ni–Vazquez–Villani (JMPA 2009) and Kotschwar–Ni (Annales Scientifiques de l'Ecole Normale Superieure 2009). Copyright © 2013 John Wiley & Sons, Ltd.

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are 2k-Einstein (in the sense that their 2k-Ricci tensor is constant) or have constant 2k-Gauss-Bonnet curvature. The results hold for a family of manifolds containing all non-flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.

In this article, we study topology of complete non-compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n -space ℝn if it contains enough rays starting from the base point. We also show that a complete non-compact n -dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ℝn if the volumes of geodesic balls in M grow properly. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal.

In this paper, we prove that complete open Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to three in which some Caffarelli– Kohn–Nirenberg type inequalities are satisfied are close to the Euclidean space.

The behaviour of the Ricci curvature along rays in a complete open manifold is examined.

First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,jR2ij [les ] r2/(n−1), where [sum ]i,jR2ij is the squared norm of the Ricci curvature tensor.

We prove that a Ricci flat end of a Riemannian manifold is asymptotically Euclidean if it is obtained from a smooth metric by a conformal inversion. A number of consequences are discussed.

We consider Riemannian orbifolds with Ricci curvature nonnegative outside a compact set and prove that the number of ends is finite. We also show that if that compact set is small then the Riemannian orbifolds have only two ends. A version of splitting theorem for orbifolds also follows as an easy consequence.

Locally symmetric KAHLER manifolds may be characterized as almost HERMITian manifolds with symplectic or holomorphic local geodesic symmetries. We extend the notion of a local geodesic symmetry and in particular, give a similar characterization of all RIEMANNian locally s-regular manifolds with an s-structure of odd order.