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

TL;DR

This paper investigates the curvature properties of the Gauss map for submanifolds with flat normal bundle in space forms, establishing a relation between the curvature of the Gauss image and the original submanifold's geometry.

Contribution

It derives a formula linking the Riemann curvature tensor of the Gauss map to the original submanifold's curvature and Weingarten operators, generalizing the theorema egregium.

Findings

The Riemann curvature tensor of the Gauss image is determined by the original submanifold's curvature.

The induced metric on the Gauss image is given by the third fundamental form.

A generalized Kulkarni-Nomizu product expresses the curvature relation.

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.