User - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:01:02Z http://mathoverflow.net/feeds/user/28053 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115415/orientation-preserving-dieomorphism-on-tn orientation preserving diﬀeomorphism on T^n unknown (google) 2012-12-04T16:16:27Z 2012-12-04T21:20:06Z <p>Given $S \in GL_n(\mathbb{R}^n)$.</p> <p>Show that $x \mapsto Sx$ is an orientation preserving diﬀeomorphism on $\mathbb{T}^n$ if and only if $S \in SL_n(\mathbb{Z}^n)$.</p> <p>I'm working on the only if part. Orientation perserving implies $\det(S) > 0$.</p> <p>How can I prove that we must have $S \in SL_n(\mathbb{Z}^n)$. I can see that when $S \in SL_n(\mathbb{R}^n)$ is not enough to be surjective, but can´t prove that $S \in SL_n(\mathbb{Z}^n)$.</p> http://mathoverflow.net/questions/112211/cotangent-space-of-the-sphere Cotangent space of the sphere unknown (google) 2012-11-12T19:24:55Z 2012-11-12T19:52:10Z <p>In analyzing the spherical pendulum the cotangent space of the sphere is defined as</p> <p>$T^*S^2 = \lbrace (q,p) \in \mathbb{R}^3 \times \mathbb{R}^3; |q| = 1, q \cdot p = 0 \rbrace$ </p> <p>my problem with this is that I see the right-hand side of the equation as a set of points, whereas I see the left-hand side as a set of linear functions on the tangent space of $S^2$. </p> <p>How can I see them as the same?</p> http://mathoverflow.net/questions/115415/orientation-preserving-dieomorphism-on-tn Comment by 2012-12-04T18:54:06Z 2012-12-04T18:54:06Z You right Andreas, my mistake, I changed it. http://mathoverflow.net/questions/115415/orientation-preserving-dieomorphism-on-tn Comment by 2012-12-04T17:34:31Z 2012-12-04T17:34:31Z I&#180;m sorry I not really follow you. I was hoping for a simple linear/algebra prove.