# Functions which are periodic along every geodesic

In an effort to understand some geometric rigidity theorems, I am curious about the following: let $$(M,g)$$ be a complete Riemannian manifold and suppose there is a nonconstant real-valued function on $$M$$ which has a period of 1 when restricted to any unit-speed geodesic of $$M$$. Does this place restrictions on $$(M,g)$$?

My feeling is that the vast majority of manifolds cannot support such a function. The round sphere clearly can.

edit: if I understand correctly- according to the article arxiv.org/abs/1511.07852 of Radeschi and Wilking (Invent. Math. 2017) given by Igor Belegradek in the comments, a theorem of Wadsley (J. Diff. Geom. 1975) shows that, for every Riemannian manifold all of whose geodesics are closed, every function on $$M$$ satisfies the above condition with some constant period; the theorem is that all lengths of closed geodesics must be multiples of a single number, when $$\pi_1(M)$$ is finite

• Does your assumption imply that all geodesics are periodic? If so there is a book by Besse "Manifolds all of whose Geodesics are Closed" which studies such manifolds. – Igor Belegradek Sep 21 at 21:46
• I'm not sure. Supposing it does, I think my condition should be significantly more restrictive, since I assume the lengths of geodesics in Besse's manifolds aren't usually all multiples of the same number – Quarto Bendir Sep 21 at 21:58
• I think the state of the art is in arxiv.org/abs/1511.07852, which is "On the Berger conjecture for manifolds all of whose geodesics are closed" by Radeschi and Wilking. – Igor Belegradek Sep 21 at 23:26
• thanks for the article, I've added a comment about it to the question – Quarto Bendir Sep 22 at 0:19

If $$f\circ \gamma$$ has period $$1$$ for all unit speed geodesics $$\gamma:\mathbb{R} \to M$$ then in particular $$f(x) = f(y)$$ whenever $$x$$ and $$y$$ are the endpoints of a geodesic segment of length $$1$$.

This implies that $$f(x) = f(y)$$ whenever there is a sequence $$x_0 = x,x_1,\ldots,x_n = y$$ such that $$x_i$$ and $$x_{i+1}$$ are the endpoints of a segment of length $$1$$ for $$i = 0,\ldots,n-1$$. Following Sunada call such a sequence a $$1$$-geodesic chain.

If any two points can be joined by a $$1$$-geodesic chain then $$f$$ would have to be constant. In a couple of papers of Sunada (e.g. Theorem C from "Mean value theorems and ergodicity of certain random walks" Compositio Mathematica 1983) conditions under which any two points can be joined by a $$1$$-geodesic chain are given.

His results imply that, in order for a non-constant $$f$$ to exist, there must be a point $$o \in M$$ such that for any unit speed geodesic with $$\gamma(0) = x$$ one has that $$\gamma(n)$$ is conjugate to $$x$$ along $$\gamma$$ for all integer $$n$$.

This implies in particular that $$M$$ is compact and its fundamental group is finite (since the condition above passes to the universal covering space as well).

For surfaces this shows (Theorem G in the aforementioned paper by Sunada) that $$M$$ would have to be either the sphere or the projective plane with a metric such that all geodesics from $$o$$ have period $$2$$.

• Thanks, very interesting! – Quarto Bendir Sep 22 at 2:01
• If $\Gamma$ is a finite subgroup of $SO(n+1)$, does there always exist a non-constant such $f$ on $S^n/\Gamma$? (Maybe this is obvious) – Kevin Casto Sep 22 at 2:42