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I think that this is treated in Helgason's Differential Geometry, Lie Groups and Symmetric Spaces. The point is that, for any Riemannian symmetric space $G/K$, one has the notion of the rank $r$ of t …

The last I checked, it was unknown whether the set of Zoll metrics on the $2$-sphere was connected. What Guillemin (V. Guillemin, The Radon transform on Zoll surfaces, Advances in Mathematics 22 (197 …

This is certainly true. If you choose $\epsilon>0$ smaller than half the injectivity radius of $g$ at $p$ (in particular, sufficiently small that the exponential map $\exp_p:T_pU\to U$ is well-define …

Now, the $n/2$-fold cover $\hat e$ of the geodesic $e$ will intersect all of the 'near-by' geodesics (which are also closed) exactly $n$ times. … (Here, 'near-by' means geodesics that stay within $\hat B\subset\Sigma$.) Q2 does not seem to me to be well-formulated. …

The answer is as follows: When $U$ is a positive-definite, symmetric, circulant matrix in $\mathrm{GL}(n,\mathbb{R})$, then there is a symmetric circulant matrix $u$ such that $U = e^u$ and the curve …

The answers to your questions about geodesics on such surfaces can be found in A. L. Besse's book Manifolds all of whose geodesics are closed (1978, Springer Ergebnisse series). … In particular, it is true that all of the geodesics that stay in the smooth part of the surface are closed, and their length is known in terms of $a$, $p$, and $q$. …

I will show that $g$ has no closed geodesics in $\mathbb{R}^2$. … Thus, $g$ has no closed geodesics in $\mathbb{R}^2$ and hence the induced metric on $\mathbb{T}$ has no closed null-homotopic geodesics. …

Now, it is easy to see that these minimal geodesics in each fixed-endpoint homotopy class cannot fall into a finite number of geodesics that are distinct in your sense. … circle of geodesics in this case. …

Using the structure equations, it is not difficult to show that, if $f:(M,g)\to(N,h)$ is a diffeomorphism of (not necessarily complete) connected surfaces that is affine in the OP's sense, i.e., $\nab …

Note: I've decided that this answer should be rearranged a bit so that it clearly separates the discussion of the basic properties of the tangent bundle from the discussion of the formulae associate …

Any surface of revolution in $3$-space with poles will have this property. The reason is that, in this case, any geodesic either goes through a pole (i.e., a point where the axis of revolution meets …