Paper ID sheet UCL-INMA-2013.01

Title

An extrinsic look at the Riemannian Hessian

Authors
P.-A. Absil, Robert Mahony, Jochen Trumpf
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.
Key words
Riemannian Hessian, Euclidean Hessian, Weingarten map, shape operator
Status
Geometric Science of Information. Lecture Notes in Computer Science, Volume 8085, 2013, pp 361-368.
Date of ID sheet creation
08 March 2013
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Errata
This errata concerns the publisher's version. The typos are corrected in the technical report available above.
  1. Page 8, first displayed equation: replace "W \dot{V}^\T V^\T \P_V = W\dot{V}^\T V" by "W \dot{V} V^\T \P_V = W\dot{V} V^\T" (transpose deleted twice and inserted once).
  2. Page 8, third displayed equation: replace "W\dot{V}^\T V + U\dot{U}^\T W" by "W\dot{V} V^\T + U\dot{U}^\T W" (transpose deleted once and inserted once).
BibTeX entry
@TECHREPORT{AbsMahTru2013.01,
 author = "P.-A. Absil and Robert Mahony and Jochen Trumpf",
 title = "An extrinsic look at the {Riemannian Hessian}",
 institution = "U.C.Louvain",
 year = 2013,
 number = "UCL-INMA-2013.01-v2",
 month = "February",
 url = "http://sites.uclouvain.be/absil/2013.01",
}
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