An Invitation to Morse Theory by Liviu Nicolaescu

Author:Liviu Nicolaescu
Language: eng
Format: epub, pdf
Publisher: Springer New York, New York, NY

(3.27)

with Hamiltonian function

Set .

This action is not effective because the subgroup

acts trivially. To obtain an effective action we need to factor out this subgroup and look at the action of . We will do this a bit later.

The Lie algebra of is identified with the subspace

The vector generates the Hamiltonian flow

with Hamiltonian function

We can now explain how to concretely factor out the action of G. This is done in two steps as follows.

Step 1. Construct a smooth surjective morphism of two-dimensional tori such that k erφ = G.

Step 2. Define a new action of on by setting

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