Hi, Let $M$ be a pseudo-Riemannian manifold and $G$ a (Lie) subgroup of $Iso(M)$ which acts on $M$ smoothly and properly. Suppose we know the orbits up diffeomorphism. Is there a systematic way to recognize the induced metric on an orbit of the action? you may assume that $M$ is a space form of constant curvature and and $G$ is compact or semisimple or ... if there is no acceptable answer in general case. thanks to all.
The orbit is homogeneous, so it's enough to compute the metric at one point, call it $x$. The map from the Lie algebra to the tangent space to the orbit at $x$ is surjective. Concretely for a vector $X$ in the Lie algebra the corresponding tangent vector is $X_M (x) := d/dt \exp (tX) \cdot x$ ( $\cdot $ denotes the action). Now take another vector $Y$ in the Lie algebra and compute the inner product between $X_M (x)$ and $Y_M (x)$. This computes the induced "metric" on $G/G_x$ ($G_x$ is the stabilizer of $x$). The word "metric" is in scary quotes because it could be zero.