Nonsmooth Optimization Techniques on Riemannian Manifolds

We present the notion of weakly metrically regular functions on manifolds. Then, a sufficient condition for a real valued function defined on a complete Riemannian manifold to be weakly metrically regular is obtained, and two optimization problems on Riemannian manifolds are considered. Moreover, we present a generalization of the Palais–Smale condition for lower semicontinuous functions defined on manifolds. Then, we use this notion to obtain necessary conditions of optimality for a general minimization problem on complete Riemannian manifolds.     

Semi-invariant Riemannian maps from almost Hermitian manifolds

We construct Gauss–Weingarten-like formulas and define O’Neill’s tensors for Riemannian maps between Riemannian manifolds. By using these new formulas, we obtain necessary and sufficient conditions for Riemannian maps to be totally geodesic. Then we introduce semi-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, give examples and investigate the geometry of leaves of the distributions defined by such maps. We also obtain necessary and sufficient conditions for semi-invariant maps to be totally geodesic and find decomposition theorems for the total manifold. Finally, we give a classification result for semi-invariant Riemannian maps with totally umbilical fibers.     

Inequalities for lower order eigenvalues of second order elliptic operators in divergence form on Riemannian manifolds

In this paper, we investigate the Dirichlet eigenvalue problems of second order elliptic operators in divergence form on bounded domains of complete Riemannian manifolds. We discuss the cases of submanifolds immersed in a Euclidean space, Riemannian manifolds admitting spherical eigenmaps, and Riemannian manifolds which admit l functions ${f_\alpha : M \longrightarrow \mathbb{R}}$ such that ${\langle \nabla f_\alpha, \nabla f_\beta \rangle = \delta_{\alpha \beta}}$ and Δf _α = 0, where ? is the gradient operator. Some inequalities for lower order eigenvalues of these problems are established. As applications of these results, we obtain some universal inequalities for lower order eigenvalues of the Dirichlet Laplacian problem. In particular, the universal inequality for eigenvalues of the Laplacian on a unit sphere is optimal.     

Convexity and Some Geometric Properties

The main goal of this paper is to present results of existence and nonexistence of convex functions on Riemannian manifolds, and in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Riemannian manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete manifolds, and we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result, we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.     

$\delta$-拼挤流形中的2-调和子流形

We study biharmonic submanifolds in δ-pinched Riemannian manifolds, and obtain some sufficient conditions for biharmonic submanifolds to be minimal ones.     

Strong Geodesic α-Preinvexity and Invariant α-Monotonicity on Riemannian Manifolds

In this article, we introduce and study a new class of generalized convex functions on Riemannian manifold, called strongly α-invex and strongly geodesic α-preinvex functions. Several kinds of invariant α-monotonicities on Riemannian manifold are introduced. We establish the relationships among the strong α-invexity, strong geodesic α-preinvexity and invariant α-monotonicities under suitable conditions. Various types of α-invexities for functions on Riemannian manifolds are introduced and relations among them are established.     

Invex sets and preinvex functions on Riemannian manifolds

The concept of a geodesic invex subset of a Riemannian manifold is introduced. Geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined. The relation between geodesic invexity and preinvexity of functions on manifolds is studied. Using proximal subdifferential, certain results concerning extremum points of a non smooth geodesic preinvex function on a geodesic invex set are obtained. The main value inequality and the mean value theorem in invexity analysis are extended to Cartan-Hadamard manifolds.     

Anti-invariant Riemannian submersions from almost Hermitian manifolds

We introduce anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a Langrangian Riemannian submersion, a special anti-invariant Riemannian submersion, to be totally geodesic. Moreover, we obtain decomposition theorems for the total manifold of such submersions.     

Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.Partially supported by NSF grant MCS77-18723(02).     

ON THE HARNACK INEQUALITY FOR HARMONIC FUNCTIONS ON COMPLETE RIEMANNIAN MAINFOLDS

First it is shown that on the complete Riemannian manifold with nonnegative Ricci curvature $\overline M$ the Sobolev type inequality $\[||\nabla u|{|_2} \geqslant {C_{n,\alpha }}||u|{|_{2\alpha }}(\alpha \geqslant 1)\]$, for all $u \in H^2_1(\overline M)$ holds if and only if $V_x(r)=Vol(B_x(r))\geq C_nr^n$ and $\alpha=\frac{n}{n-2}$. Let M be a complete Riemannian manifolds which is uniformly equivalent to $\overline M$, and assume that $V_x(r)\geq C_nr^n$ on $\overline M$. Then it is prioved that the John-Nirenberg inequality, holds on M. Finally, based on the Sobolev inequality and John-Nirenberg inequality, the Harnack inequality for harmonic functions on M is obtained by the method of Moser, arid consequently some Liouville theorems for harmonic functions and harmonic maps on M are proved.     

Harmonic morphisms as unit normal bundles¶of minimal surfaces

Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres. Received: 6 February 1999     

Integrable Geodesic Flows on Riemannian Manifolds

We discuss the notion of geodesics and study the global behavior of geodesics on closed Riemannian manifolds. In particular, we emphasize the case of so-called integrable geodesic flows.     

Semi-slant Riemannian map

As a generalization of slant submersions [18], semi-slant submersions [15], and slant Riemannian maps [21], we define the notion of semi-slant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We study the integrability of distributions, the geometry of fibers, the harmonicity of such maps, etc. We also find a condition for such maps to be totally geodesic and investigate some decomposition theorems. Moreover, we give examples.     

Geodesic vector fields and Eikonal equation on a Riemannian manifold

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.     

A NOTE ON THE NULLITY OF HOMOTOPY GROUP FOR COMPLETE THREE DIMENSIONAL MANIFOLDS WITH RICCI ≧ 0

This note shows the nullity of homotopy groups for complete three dimensional manifoldswith Ricci≥ 0 under some growth condition of the geodesic ball. The author also gives someexamples which show the growth condition here is optimal in some sense.     

Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on $\mathsf{C }^{\infty }_0$ , and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli–Dirac operators that describe the energy of Hydrogen type atoms on Riemannian $3$ -manifolds.     

A NOTE ON THE NULLITY OF HOMOTOPY GROUP FOR COMPLETE THREE DIMENSIONAL MANIFOLDS WITH RICCI ≧ 0

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＭ３ｂｅａｃｏｍｐｌｅｔｅ，ｎｏｎｃｏｍｐａｃｔＲｉｅｍａｎｎｉａｎｍａｎｉｆｏｌｄ，ａｎｄＲｉｃｃｉＭｂｅｉｔｓＲｉｃｉｃｕｒｖａｔｕｒｅ．ＳｃｈｏｅｎＹａｕ［１］ｐｒｏｖｅｄｔｈｅｆｏｌｏｗｉｎｇＴｈｅｏｒ...     

Pseudo-Riemannian weakly symmetric manifolds

There is a well-developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit property, weak symmetries, and the structure of the nilradical of the isometry group. Also, we give a number of examples of weakly symmetric pseudo-Riemannian manifolds, some mirroring the Riemannian case and some indicating the problems in extending Riemannian results to weakly symmetric pseudo-Riemannian spaces.     

Generalized ($\kappa,\mu $)-contact metric manifolds with $\|{grad}\kappa \| =$ constant

We locally classify the 3-dimensional generalized ($\kappa,\mu$)-contact metric manifolds, which satisfy the condition $\Vert grad\kappa \Vert =$const. ($\not=0$). This class of manifolds is determined by two arbitrary functions of one variable.     

Weakly Disk-Homogeneous Spaces

Naturally reductive Riemannian homogeneous spaces and more generally, g.o. spaces, have the property that the volume of a geodesic disk normal to a geodesic and with center on that geodesic remains constant when the center moves along that central geodesic. Riemannian manifolds having that property for arbitrary geodesics and all sufficiently small geodesic disks are called weakly disk-homogeneous. Since, up to local isometries, there are no other examples known for dimension > 2, we investigate whether a possible converse holds or not. Besides some general results, we give a positive answer for three-dimensional and for several classes of four-dimensional manifolds. Some related results are discussed, in particular about four-dimensional Einstein C-spaces.