Abstract
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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Absil, P.A., Mahony, R., Trumpf, J. (2013). An Extrinsic Look at the Riemannian Hessian. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_39
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DOI: https://doi.org/10.1007/978-3-642-40020-9_39
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