Paper ID sheet UCL-INMA-2013.01


An extrinsic look at the Riemannian Hessian

P.-A. Absil, Robert Mahony, Jochen Trumpf
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
Geometric Science of Information. Lecture Notes in Computer Science, Volume 8085, 2013, pp 361-368
BibTeX entry

 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 = "",
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).