## More on the Morse-Bott lemma and fixed-point sets of conformal flows

### Time

Apr 10 2012 - 2:30pm - 3:30 pm

MW 154

### Speaker

Andrzej Derdzinski (OSU)

### Abstract

Let Z be the zero set of a conformal vector field v (that is, the fixed-point set of a conformal flow) on a pseudo-Riemannian manifold of dimension n>2. For points x of Z which are nonessential, in the sense that some local conformal change of the metric at x turns the flow into one consisting of isometries, the structure of Z near x is completely understood (a result of Kobayashi, 1958). The remaining case is described by the following

THEOREM. If a point x of Z is essential, then the exponential mapping at x diffeomorphically identifies a neighborhood of x in Z with a neighborhood of 0 in the null cone of a specific subspace of the tangent space at x.

The proof uses the fact that - aside from some simpler special cases - Z, near x, coincides with the zero set of a certain function f defined on a submanifold K containing x, while the hypotheses of the Morse-Bott lemma are satisfied both by f and by its restriction to a suitable codimension-one submanifold of K.
Last updated by derdzinski.1 on 04/06/12