What is the order of differentiability of the tangent bundle of a C^2- manifold? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:49:10Z http://mathoverflow.net/feeds/question/25003 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25003/what-is-the-order-of-differentiability-of-the-tangent-bundle-of-a-c2-manifold What is the order of differentiability of the tangent bundle of a C^2- manifold? Akela 2010-05-17T12:56:59Z 2010-05-17T13:19:47Z <p>The construction of tangent bundle on a $C^\infty$ manifold, as for example in the book of Warner, uses the existence of double derivatives. Of course the tangent space for a point is first constructed in case of an open set in a Euclidean space and then the whole setup is glued up. But then in the neighborhood of a point the second derivatives might be different for a different $C^1$-chart around the point. So I suppose the tangent space/tangent bundle construction is for $C^2$-manifolds. </p> <p>Question is:</p> <blockquote> <p>Since $C^2$ is apparently the minimum condition for existence of tangent bundle, is the tangent bundle just $C^1$, or it is again $C^2$?</p> </blockquote> http://mathoverflow.net/questions/25003/what-is-the-order-of-differentiability-of-the-tangent-bundle-of-a-c2-manifold/25005#25005 Answer by AndrĂ© Henriques for What is the order of differentiability of the tangent bundle of a C^2- manifold? AndrĂ© Henriques 2010-05-17T13:19:47Z 2010-05-17T13:19:47Z <p>The tangent bundle of a $C^1$ manifold exists: it's a $C^0$ manifold.<br> Similarly, for every $n$, the tangent bundle of a $C^n$ manifold is a $C^{n-1}$ manifold.</p>