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.
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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. |
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