Research on curvature structures and topological structures of submanifolds in Riemannian manifolds

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In this project, we mainly investigated the curvature structures and topological structures of submanifolds in Riemannian manifolds and the geometry of eigenvalues and eigenfunctions of Laplacian on Riemannian manifolds. It is our purpose to research geometric problems on properties oftopology and curvatures of many kinds of manifolds by means of many different methods. We studied (1)the geometry of topological structures and curvature structures of submanifolds in Euclidean spaces and spheres. (2)the geometry of compact hypersurfaces with infinite fundamental group in spheres. (3)the geometry of curvature structures of complete hypersurfaces in 4-dimensional space forms. (4)the geometry of curvature structures of complete space-like hypersurfaces in Lorentz spaces. (5)the geometry ofeigenvalues and eigenfunctions of Laplacian on Riemannian manifolds. (6)the geometry of sphere theorems. (7)the geometry of conformally projective structures of satistical manifolds. (8)the geometry on radial curvature and topology of manifolds and obtained many important results.It is characterizations of our project that important contributions on the research of topological structures and curvature structures of complete submanifolds in Riemannian manifolds are obtained. Furthermore, Cheng and Yang obtained important results for the investigation on eigenvalues of Laplacian on Riemannian manifolds.