Riemannian submanifold with extreme value of sectional curvature

Abstract

We studies a Riemannian manifold which has an extreme valud of sectional curvature. In this thesis, using the geodesic variation, we constructed a 2-demensional totally geodesic submanifold in M when M has bounded sectional curvature either above or below. We showed that the sectional curvature of M takes the extreme value over the surface $\Sigma$ if and only if $\Sigma$ is locally isometric to space form and totally geodesic as an immersed submanifold.