# Group Action on a Space

### by bob jantzen [2001]

17 pages of notes for an independent study by Christopher Pilman, in addition
to the previous sets of differential geometry notes [1984,
1991], with several pages on the Lie derivative copied
from Introduction to Cosmological Models 2.

The rotations in the plane are used to motivate a general 1-parameter group
of transformations, using the exponential map, then specializing back to the
rotations showing the relation to the matrix generators and matrix
transformation. Comoving coordinates are found for this example. Then dragging
and the Lie derivative are introduced. Then the relationship between the
rotation generators and linear and angular momentum is discussed. Next
r-parameter groups and rotations of space, then boosts of 2-d Minkowski
spacetime, and the generators of the 4-d Lorentz group. Finally the Lie
derivative and isometry actions are touched upon, with a final exercise.