An Extrinsic Look at the Riemannian Hessian

Let f be a real-valued function on a Riemannian submanifold of a Euclidean space, and let \(\bar{f}\) be a local extension of f. We show that the Riemannian Hessian of f can be conveniently obtained from the Euclidean gradient and Hessian of \(\bar{f}\) by means of two manifold-specific objects: the orthogonal projector onto the tangent space and the Weingarten map. Expressions for the Weingarten map are provided on various specific submanifolds.

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