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?(adsbygoogle = window.adsbygoogle || []).push({});

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# Moment maps and Morse functions

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