# Moment maps and Morse functions

1. Jan 16, 2016

### Ssnow

It is know that let $M$ a compact symplectic manifold with $G=T^{d_{T}}$ a torus of dimension $d_{T}$ acting on $M$ in Hamiltonian fashion with Moment map $\Phi:M\rightarrow \mathfrak{t}^{*}$, then $\Phi^{\xi}=\langle \Phi(m),\xi\rangle$ is a Morse function in each of its component (for $\xi\in\mathfrak{t}$). What I want to discuss here is what happen if we consider now the product group $P=G\times T^{d_{T}}$ where $G$ is a Lie group of dimension $d_{G}$, assuming that $P$ acts in Hamiltonian way and that at each point of $M$ the moment map of $G$ is $\Phi_{G}(m)=0\in \mathfrak{g}^{*}$. It is $\Phi_{P}$ a Morse function?

2. Jan 21, 2016