Principal curvatures and curvature directions - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T19:08:07Z http://mathoverflow.net/feeds/question/49946 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49946/principal-curvatures-and-curvature-directions Principal curvatures and curvature directions Dror Atariah 2010-12-20T09:51:05Z 2010-12-20T17:15:25Z <p>Last week I considered again <em>principal curvature (pc)</em> and <em>principal curvature directions (pcd)</em> of a, for the sake of simplicity, 2-manifold embedded in 3-space. In this simple case, the pc and pcd of at a point are the eigenvalues and eigenvectors of the shape operator. The magnitude of the pc's corresponds to the minimal and maximal normal curvature at the point. My question, however, is:</p> <blockquote> <p>What does the principal curvatures direction <strong>magnitude</strong> represents?</p> </blockquote> <p>In the textbooks I looked up in (Kühnel and do Camro) I couldn't find a reference to the principal curvature direction's magnitude. Is there something known about this? Is it something basic (maybe even from linear algebra)?</p> <p>Edit 1: A somewhat more general, but related, question is:</p> <blockquote> <p>What is the geometrical meaning of an eigenvector's magnitude?</p> </blockquote> http://mathoverflow.net/questions/49946/principal-curvatures-and-curvature-directions/49951#49951 Answer by Zhen Lin for Principal curvatures and curvature directions Zhen Lin 2010-12-20T10:50:06Z 2010-12-20T10:50:06Z <p>Unless there's an additional constraint on the defintion of the principal curvature directions, being just defined as eigenvectors means their magnitude is arbitrary and so meaningless.</p> http://mathoverflow.net/questions/49946/principal-curvatures-and-curvature-directions/49974#49974 Answer by Dick Palais for Principal curvatures and curvature directions Dick Palais 2010-12-20T16:27:35Z 2010-12-20T16:27:35Z <p>While all of the above remarks about the principal curvature magnitudes being arbitrary are correct, I think it is fairly customary for differential geometers to think of the eigenvectors of the shape operator as being the principal axes of the the curvature ellipsoid, and so giving them lengths equal to the principal curvatures. Of course this only makes sense on the complement of the umbilic points.(Recall that the umbilics are the points where the principal curvatures are equal, so that at these points the principal directions are not well-defined.) </p> http://mathoverflow.net/questions/49946/principal-curvatures-and-curvature-directions/49978#49978 Answer by John Bentin for Principal curvatures and curvature directions John Bentin 2010-12-20T17:15:25Z 2010-12-20T17:15:25Z <p>In general, the concept of "eigenvector" is slightly misleading. The fundamental concept is <em>eigenspace</em>. However, eigenspaces of dimension greater than 1 are generally considered pathological and rather a nuisance. Mostly we would rather have one-dimensional eigenspaces, and for these any convenient nonzero element---an eigenvector--- will serve as a representative. It would be pedantically correct but encumbering to insist on talking about one-dimensional eigenspaces rather than eigenvectors. The arbitrariness of eigenvectors becomes clearer when we really have to deal with a multidimensional eigenspace.</p>