Some results in geometry of hypersurfaces

Abstract

OKAYASU Introduction.In this paper we get several theorems about hypersurfaces in space forms.In section 1, we show that if x : M n -*E n+1 is an isometric immersion of an n-dimensional complete non-compact Riemannian manifold whose sectional curvatures are greater than or equal to 0, then x(M) is unbounded in E n+1 .We can prove this using Sacksteder theorem [12] which states that under the above condition x(M) is the boundary of a convex body in E n+1 .But his proof is rather long and his theorem is more than what we need.do.Carmo and Lima [3] gave an independent proof of Sacksteder theorem, but it is also long.So we give a direct and easy proof using so-called Beltrami maps which are defined in do.Carmo and Warner [4].In section 2, we show that if x : M n ->S n+1 (l) is an isometric immersion of an n-dimensional complete Riemannian manifold whose sectional curvatures are less than or equal to 1 and n is greater than 3, then x(M) is totally geodesic.Ferus almost proved this result in [6], [7].We consider higher codimensional cases.All manifolds we consider in this paper are class C°°, connected and have dimensions greater than or equal to 2. All immersions and vector fields are C°°.

Locations

• Kodai Mathematical Journal - View
• Project Euclid (Cornell University) - View - PDF