- #1

- 3,506

- 26

## Main Question or Discussion Point

Vector fields generate flows, i.e. one-parameter groups of diffeomorphisms, which are profusely used in physics from the streamlines of velocity flows in fluid dynamics to currents as flows of charge in electromagnetism, and when the flows preserve the metric we talk about Killing vector fields and isometries are defined and used i.e. in GR.

How exactly is this generalized to the case of tensor fields?, let's concentrate on rank-2 tensor fields that are most often found in physics. Intuitively one would think they would generate two-parameters groups of diffeomorphisms?. Say for instance if the tensor field is a metric tensor(thinking about Klein's Erlangen program), they would generate isometries in the form of transformations like translations, rotations, reflections involving two directions? For other two-rank tensors that have physical significance , say the stress-energy tensor, what kind of (local)"flow" would it generate?

How exactly is this generalized to the case of tensor fields?, let's concentrate on rank-2 tensor fields that are most often found in physics. Intuitively one would think they would generate two-parameters groups of diffeomorphisms?. Say for instance if the tensor field is a metric tensor(thinking about Klein's Erlangen program), they would generate isometries in the form of transformations like translations, rotations, reflections involving two directions? For other two-rank tensors that have physical significance , say the stress-energy tensor, what kind of (local)"flow" would it generate?

Last edited: