Curvature of the Gauss map for normally flat submanifolds in space forms

Abstract

For a submanifold with flat normal bundle in a space form there is a normal orthonormal basis that simultaneously diagonalizes the corresponding Weingarten operators, and at which these operators satisfy a simple Codazzi symmetry. When the second fundamental form has zero index of relative nullity, the image of the Gauss map as defined by Obata is a submanifold of a generalized Grassmannian manifold, with induced metric given by the third fundamental form, corresponding to the sum of the squares of these operators. In this case, we show that the Riemann curvature tensor of the Gauss image is completely determined by the curvature and Weingarten operators of the original submanifold, in a form analogous to a theorema egregium that relates the intrinsic geometry of the Gauss map with the extrinsic geometry of the original embedding. For this, we derive the difference vector of the induced Levi-Civita connections and express the Riemann curvature tensor with a generalized Kulkarni-Nomizu product.