I've just started reading a paper called "Formules de Localisation et Formules de Paul Lévy" by Bismut, and I didn't understand two things :
1). Let $M$ be a compact connected oriented manifold, suppose that $TM$ is endowed whith a metric $g$. Let $X$ be a killing vector field. Let $F$ be the manifold of zeros of $X$. Let $N$ be the normal bundle of $F$ in M. The author said that the Levi-cevita connection $\nabla$ of $M$ induced a euclidean connection $\nabla^N$ in $N$.
Why this true: How one construct this connection ?
- And then he said: Let $J^X$ be the infinitesimal action of $X$ on N. In local coordinates, $J^X$ is given by: $$J^X = \partial X / \partial X.$$
This is the first time I heard of this notion of infinitesimal action of a vector field on a manifold ! All I've found on internet was about the notion of infitesimal action of a vector field on a manifold wich is provided with a Lie group .
Any help would be greatly appreciated.
Thanks!