Compute differential on cotangent bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:05:29Z http://mathoverflow.net/feeds/question/73808 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73808/compute-differential-on-cotangent-bundle Compute differential on cotangent bundle unknown (google) 2011-08-26T23:28:01Z 2011-08-27T08:05:47Z <p>Hi,</p> <p>This is my question. Can we compute easily the differential of the following map ?</p> <p>$$f:(x,\xi^\star)\in TS^{2n-1} \mapsto \xi^\star(ix)\in \mathbb{R}$$</p> <p>where $TS^{2n-1}$ is the cotangent bundle of the odd sphere $S^{2n-1}\subset \mathbb{C}^n$. Notice that $f$ is well defined because if $x\in S^{2n-1}\subset \mathbb{C}^n$ then $ix\in T_x S^{2n-1}$.</p> <p>Thanks</p> http://mathoverflow.net/questions/73808/compute-differential-on-cotangent-bundle/73827#73827 Answer by Ben McKay for Compute differential on cotangent bundle Ben McKay 2011-08-27T08:05:47Z 2011-08-27T08:05:47Z <p>The map is the restriction to $TS^{2n-1}$ of a quadratic map, so its differential is the restriction of the differential of that map. In coordinates $z^{\mu}$ on $\mathbb{C}^n$, we get coordinates $z^{\mu},\xi_{\mu}$ on $T^*\mathbb{C}^n$, and $z^{\mu},\xi^{\mu}$ on $T\mathbb{C}^n$. The function is $f=\left&lt;\xi, \sqrt{-1} z\right>=-\sqrt{-1}\xi^{\mu} z^{\bar{\mu}} +\sqrt{-1} \xi^{\bar{\mu}} z^{\mu}$, so has differential $df=-\sqrt{-1}\left(\xi^{\mu} dz^{\bar{\mu}} + z^{\bar{\mu}} d\xi^{\mu}\right)+\sqrt{-1}\left(\xi^{\bar\mu} dz^{\mu} + z^{\mu} d\xi^{\bar\mu}\right)$. The equations on the sphere are $z^{\mu} z^{\bar{\mu}} = 1$ so the tangent bundle of the sphere is $z^{\mu} \xi^{\bar{\mu}} + \xi^{\mu} z^{\bar{\mu}} = 0$. Taking exterior derivative, you can simplify $df$ a little.</p>