Actions on Parametric Representations

Abstract

Now with the use of parametric representations, we come to the basic idea of this work, namely we can define the arbitrary nonlinear Lie group actions on functions

$$ G\;x\;C_{_n}^\infty \left( M \right) \to C_{_n}^\infty \left( M \right) $$

((4.1))

in the following simple and natural way. Given g ∈ G and a function $\Lambda M$ from C ^∞ _n (M), we define

$$ gV = goV $$

((4.2))

where in the right hand term, g is the mapping in (1.5). In other words, we use as definition of the Lie group action the commutative diagram

Clearly, with the definition (4.2), (4.3), we have

$$ gV \in C_n^\infty \left( M \right),\;\Lambda M $$

((4.4))

that is, gV and V have the same domain of definition ∧, and the same range M.