tangent sphere bundle over sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:09:27Z http://mathoverflow.net/feeds/question/25659 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25659/tangent-sphere-bundle-over-sphere tangent sphere bundle over sphere sara 2010-05-23T12:02:32Z 2010-06-02T17:11:53Z <p>are there some general description about tangent sphere bundle over sphere? (it is a special $S^{n-1}$bundle over $S^n$)</p> <p>say for n=1,it is trivial,$S^0\times S^1$,for n=2,it is $SO(3)\cong \mathbb{R}P^3$, for n=3,it is trivial again,so it is for n=7. how about other cases.</p> http://mathoverflow.net/questions/25659/tangent-sphere-bundle-over-sphere/25662#25662 Answer by Charles Matthews for tangent sphere bundle over sphere Charles Matthews 2010-05-23T13:14:26Z 2010-05-23T13:14:26Z <p>It's not so simple in general: see the "vector fields on spheres" problem at <a href="http://en.wikipedia.org/wiki/Vector_fields_on_spheres" rel="nofollow">http://en.wikipedia.org/wiki/Vector_fields_on_spheres</a> . Odd and even dimensions are different in nature because of the Euler characteristic.</p> http://mathoverflow.net/questions/25659/tangent-sphere-bundle-over-sphere/26842#26842 Answer by Ryan Budney for tangent sphere bundle over sphere Ryan Budney 2010-06-02T16:56:50Z 2010-06-02T17:11:53Z <p>If you like clutching maps descriptions of bundles the sphere has a nice one. Think of $S^n$ as the union of two discs corresponding to an upper and lower hemi-sphere. Then the tangent bundle trivializes over both hemispheres. You can write down the trivializations explicitly with some linear algebra constructions. Think of the intersection of the two hemi-spheres as an $S^{n-1}$, this allows you to think of the tangent bundle as a union $D^n \times \mathbb R^n \cup D^n \times \mathbb R^n$ along the common boundary $S^{n-1} \times \mathbb R^n$. The clutching (gluing) map is then a map of the form:</p> <p>$$c: S^{n-1} \to SO_n$$</p> <p>and it is explicitly the map $c(v) = M(v)M(x_{n+1})$</p> <p>where $v \in S^{n-1} \subset \mathbb R^n \subset \mathbb R^{n+1}$ where we think of $\mathbb R^n$ as the orthogonal complement of the $(n+1)$-st coordinate vector $x_{n+1}$ in $\mathbb R^{n+1}$. $M(v)$ denotes mirror reflection fixing the orthogonal complement to $v$. </p> <p>The basic idea in this construction is that if one takes a geodesic between two points on a sphere, parallel transport from one point to the other can be written as a composite of two reflections, the latter reflection corresponding to the mid-point of the geodesic, the initial reflection corresponding to the initial point of the geodesic. </p>