Showing papers on "Unit tangent bundle published in 1985"

A characterization of seven compact Kaehler submanifolds by holomorphic pinching

Abstract: From the results of Simons [9] and Chern, Do Carmo and Kobayashi [2], we know that, in the class of compact minimal submanifolds of a sphere, the totally geodesic submanifolds are isolated and that some simple minimal submanifolds can be characterized by suitable pinching on their curvatures. These ideas are extended naturally to compact Kaehler submanifolds of a complex projective space. The problem of the best pinching for the above submanifolds was studied later by several authors, e.g. Yau [11] and Ogiue [4]. For surfaces and hypersurfaces, the problem is completely resolved. However in the general case we have only partial results. Let M' be a compact Kaehler submanifold, of complex dimension n, immersed in the complex projective space CPtm(1) endowed with the Fubini-Study metric of constant holomorphic sectional curvature 1. Let H and K be the holomorphic sectional curvature and the sectional curvature of Mn respectively. Ogiue conjectured the following: (1) If H > , or (2) If n ? 2 and K> , or (3) If m-n n(n + ) and K > 0, 2 then Mn is a linear subvariety of CPm(1). Recently, using natural arguments at the minimum of the function H defined on the unit tangent bundle of MW, the author [7] and Verstraelen and the author [8] resolved the conjectures (1) and (2) respectively. In this paper we obtain the following complete solution of the pinching problem in the Kaehlerian case.

Singular points of curve families on surfaces

Abstract: l Introdnction It is well known that there are two cannonical bilinear forms on the tangent bundle of a smooth oriented surface which is immersed in the 3dimensional Euclidian space These are called the first fundamental form and the second fundamental form The principal curvature of the surface are defined by comparing these two forms And a point where two principal curvatures coinside is called a umbilic point Except for umblic points there exists a decomposition of the tangent space into two direct summands, each of which is tangent to the one of principal curvatures And if a curve tangents t6 those tangent lines of direct summands at any points of it, it is called a curvature curve In general for a given symmetric bilinear form on a 2-dimensional Riemannian manifold, we define the corresponding curve fammily with singularities and its local topological classification in the section 10 And we show the structurally stable condition of curve ･families on 2-dimensional closed Riemannian manifolds in the section 15