Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Rigidity of spheres in Riemannian manifolds and a non-embedding theorem

Letf:M be an isometric immersion between Riemannian manifolds. The purpose of this paper is to find the minimum possible conditions onM and (in the terms of curvatures and external diameter) in order to the image off be contained in a sphere. Our results generalize the other authors work in three major steps, domain, range and the codimension of immersions. As a byproduct, we obtain the non-embedding theorems Chern-Kuiper, Moore and Jacobowitz. The proofs are based on the maximum (comparison) principle.The author was supported by FAPEMIG Grant CEX-946/98.     

Isometry groups of Riemannian fibrations

The second author gratefully acknowledges support for the research in this project by a grant from the University of Oklahoma.     

The Isometry of Riemannian Manifold to a Sphere

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完备Riemann流形与球面的等距

研究Riemann流形的球面特征是一个颇有兴趣的问题,特别是考虑完备Riemann流形M在什么条件下与一球面等距.为此,Obata曾得到两个微分方程组,证明它们在M上非常数解的存在性等价于M与一个球面等距,其中一个方程组解的存在与共形向量场的存在有关.人们由此给出M在紧致情况下很多解的存在条件(如[3]).而另一个是下面的(也见[4]).定理A设M为n维完备、连通、单连通的Riemann流形,则下列微分方程组     

Transitive Isometry Groups of Aspheric Riemannian Manifolds

We consider the isometry groups of Riemannian solvmanifolds and also study a wider class of homogeneous aspheric Riemannian spaces. We clarify the topological structure of these spaces (Theorem 1). We demonstrate that each Riemannian space with a maximally symmetric metric admits an almost simply transitive action of a Lie group with triangular radical (Theorem 2). We apply this result to studying the isometry groups of solvmanifolds and, in particular, solvable Lie groups with some invariant Riemannian metric.     

The existence of harmonic maps from Finsler manifolds to Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact Finsler manifold M to a com-pact Riemannian manifold with non-positive sectional curvature can be deformed into aharmonic map which has minimum energy in its homotopy class.     

Knots in Riemannian manifolds

In this paper we study submanifolds with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if ${K\subset (S^n, g)}$ is a totally geodesic submanifold of codimension 2 in a Riemannian sphere with positive sectional curvature where n ≥ 5, then K is homeomorphic to S ^n–2 and the fundamental group of the knot complement ${\pi _1(S^n-K)\cong \mathbb{Z}}$ .     

Liouville-type theorems for CC-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.     

Levi-parallel contact Riemannian manifolds

We study the Riemannian geometry of contact manifolds with respect to a fixed admissible metric, making the Reeb vector field unitary and orthogonal to the contact distribution, under the assumption that the Levi–Tanaka form is parallel with respect to a canonical connection with torsion.     

Flowers on Riemannian manifolds

In this paper, we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let M ⁿ be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that consists of one vertex and at most 2n ? 1 geodesic loops based at that vertex of total length ≤ 2n!d, where d is the diameter of M ⁿ . We also show that there exists a minimal geodesic net that consists of one vertex and at most ${3^{(n+1)^2}}$ loops of total length ${\leq2 (n+1)!^2 3^{(n+1)^3}\,Fill\,Rad\,M^n \leq2(n+1)!^{\frac{5}{2}}3^{(n+1)^3}(n+1)n^n vol(M^n)^{\frac{1}{n}}}$ , where Fill Rad M ⁿ denotes the filling radius and vol(M ⁿ ) denotes the volume of M ⁿ .