2
$\begingroup$

Let $X$ be a vector field on a manifold $M$ of dimension $m$ (compact, connected, oriented) with isolated zero $z$. Let $B$ be a ball around $z$ such that $X$ has no other zeroes in $B$. We define the index of $X$ at $z$ is the degree of the map $f$ from boundary of $B$ to $S^{m-1}$ such that $f$ maps $y$ to $\frac{X(y)}{|X(y)|}$. I need to prove that this definition is well-defined (may be for the easy case $M$ is a submanifold of $\mathbb{R}^k$).

  • 0
    For some basic information about writing math at this site see e.g. [here](http://meta.math.stackexchange.com/questions/5020/), [here](http://meta.stackexchange.com/a/70559/155238), [here](http://meta.math.stackexchange.com/questions/1773/) and [here](http://math.stackexchange.com/editing-help#latex).2012-10-03

0 Answers 0