What is the Role of Lie Groups in Isometry Actions on Spaces?

[WWGD]

Hi, everyone:

I am asked to show that a group G acts by isometries on a space X.

I am not clear about the languange, does anyone know what this means?.

Do I need to show that the action preserves distance, i.e, that

d(x,y)=d(gx,gy)?.

Thanks.

 

[DeadWolfe]

Depends on context, but if it is a Riemannian manifold, presumably you want to show that the differential of the action preserves the metric tensor.

 

[WWGD]

Thanks. I am only told of a linear group, i.e, a group of matrices
acting on S^3. There is a question on geodesics, so you may
be right, and we may need to consider this as a Riemannian mfld.