Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.