A result of Helgason's is that any self-intersecting geodesic in a Riemannian globally symmetric space is simple and closed. To what extent does this generalise to Riemannian naturally reductive homogeneous spaces with the canonical connection?
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3 Answers
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Every geodesic loop on a compact homogeneous space is a closed geodesic. Let c be such a loop c(L)=c(0). Take (n-1) Killing vector fields $X_i$ whose value at p are a basis of $c'(0)^\perp$. Then $X_i$ restricted to c is a Jacobi field and hence $<X_i(c(t),c'(t)>$ is linear. Since X has bounded length, $<X_i(c(t),c'(t)>=0$ and hence $c'(L)=c'(0)$. I think this may be also true if M is not compact.
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For a homogeneous metric there is no reason for a geodesic loop at a point to be necessarily a closed geodesic. Counterexamples can probably be constructed already using metrics on the 3-sphere constructed by rescaling the round metric along the fiber of the Hopf fibration.
answered Dec 10, 2015 at 7:52
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$\begingroup$ Thanks for the answer, Katz. However, what I really want to know is whether every such homogeneous space, which is not symmetric, admits a self-intersecting geodesic. $\endgroup$ Commented Dec 10, 2015 at 8:24
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$\begingroup$ No, not every homogeneous space will have a self-intersecting geodesic. Take for example your favorite simply connected space of nonpositive curvature. You can certainly pick one that's not symmetric. $\endgroup$ Commented Dec 10, 2015 at 8:52
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$\begingroup$ Sorry, I should have said that the space must have positive curvature. Are there any examples in this class? $\endgroup$ Commented Dec 10, 2015 at 9:16
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It turns out that Helgason's result does extend to naturally reductive homogeneous spaces. One reference for this is Wolfang Ziller's paper "Closed geodesics on homogeneous spaces", Math. Z. 152, 1976.