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For a submanifold with flat normal bundle in a space form there is a normalorthonormal basis that simultaneously diagonalizes the corresponding Weingartenoperators, and at which these operators satisfy a simple Codazzi symmetry. Whenthe second fundamental form has zero index of relative nullity, the image ofthe Gauss map as defined by Obata is a submanifold of a generalizedGrassmannian manifold, with induced metric given by the third fundamental form,corresponding to the sum of the squares of these operators. In this case, weshow that the Riemann curvature tensor of the Gauss image is completelydetermined by the curvature and Weingarten operators of the originalsubmanifold, in a form analogous to a theorema egregium that relates theintrinsic geometry of the Gauss map with the extrinsic geometry of the originalembedding. For this, we derive the difference vector of the induced Levi-Civitaconnections and express the Riemann curvature tensor with a generalizedKulkarni-Nomizu product.

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