Riemannian structure on a unit sphere of a Banach space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:59:57Z http://mathoverflow.net/feeds/question/46692 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46692/riemannian-structure-on-a-unit-sphere-of-a-banach-space Riemannian structure on a unit sphere of a Banach space Dima 2010-11-19T23:01:26Z 2011-03-12T21:22:14Z <p>For a finite dimensional Banach space $V$ with strictly convex and smooth norm (outside the origin), there is a natural way to induce a Riemannian structure on the unit sphere $S$ (see Alvarez-Paiva, "Some problems on Finsler Geometry"): consider $L=\|x\|^2/2$, then $dL$ is a diffeomorphism from $V\backslash0$ to its dual $V^*\backslash0$. Now, for $q\in S$ we get $D_q(dL):T_qV \rightarrow T_qV^*$, and for $u,v\in T_qS$ take $g_q(u,v)=D_q(dL)(u)(v)$. What is known about this structure? What are the geodesics, for instance? What are their lengths?</p> http://mathoverflow.net/questions/46692/riemannian-structure-on-a-unit-sphere-of-a-banach-space/46722#46722 Answer by Sky for Riemannian structure on a unit sphere of a Banach space Sky 2010-11-20T09:48:37Z 2010-11-20T15:21:27Z <p>Maybe i'm wrong, but isn't $D_q(dL)$ the legendre transform of of the "Lagrangian"? In which case you should get the usual metric back. See for example Sternberg's book on differential geometry (the chapter on calculus of variations) or (i know, it sounds strange but) Mackey's Mathematical Foundations of Quantum Mechanics (the first chapter). If my memory doesn't fail me Abraham, Marsden should have something on that specific example too.</p> <p>Edit: This applies in the common case where the norm is induced by an inner product.</p>