sometimes also denoted , is called the fundamental vector field corresponding to . If is a curve around representing , then is represented by .
Analogously, one can define the fundamental vector field for the right actions.
There is a dual notion as well. Given a right -differentiable -manifold , a -differentiable -form with values in ,
is called the fundamental differential form corresponding to , if for all , .
The fundamental vector fields are important in study of differentiable actions and particularly useful in the basic study of Ehresmann connections. Indeed, the connection form is a fundamental form (“1st Ehresmann condition”), and there is a characterization of those fundamental forms which are connection forms (“2nd Ehresmann condition”).
Revised on December 7, 2011 02:12:32
by Toby Bartels