22

Edit: See the end for a summary of this answer I disagree with the statement "One can construct an appropriate surface patch locally in a neighborhood of each point". In fact, there are local obstructions to the existence of the desired surface. Let $\gamma:(-\epsilon,\epsilon) \rightarrow \mathbb{R}^3$ be an embedded arc parameterized proportional to arc ...


18

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 associated to a connection. The content is the same, but I hope it's clearer. The standard way to discuss the geometry of connections and geodesic flow 'invariantly' (...


13

Yes, there is at least one simple closed geodesic for every smooth metric on $\mathbb{S}^n.$ This is a result of G. D. Birckhoff (1927) - ("Dynamical Systems, AMS Coll. Pub. vol 9). It was shown by Lyusternik that there were at least $n$ closed geodesics on $\mathbb{S}^n,$ and the sharp result (Alber-Klingenberg) is that there are $2n-s - 1,$ where $s = n - ...


12

Any smooth compact surface smoothly embedded in $\mathbb{R}^3$ that is not the $2$-sphere must have an infinite fundamental group and hence must have infinitely many distinct (in your sense) geodesics joining any two distinct points. This result follows from Morse theory: If $S$ is the surface and $a$ and $b$ are points on it, then each fixed-endpoint ...


12

For $\mathbb{R}^n$: the fundamental theorem of projective geometry (proof: https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $\mathbb{R}^n$ taking lines to lines are the affine maps $x\mapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$. For $S^n$: a theorem by the same name shows that the bijections of ...


11

The geodesics are straight lines, in geodesic normal coordinates, just when the associated projective connection is flat. See Kobayashi and Nagano, On projective connections, Journal of Mathematics and Mechanics, vol. 13, no. 2, 1964. If an affine connection is projectively flat, then the Weyl and Cotton tensors vanish, as these are projective connection ...


10

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 $\gamma(t) = e^{tu}$ (which is a positive-definite, symmetric, circulant matrix for all $t$) is a geodesic in the metric $g_\alpha$ on $\mathrm{GL}(n,\mathbb{R})...


10

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). Especially, you should have a look at Chapter 4 on surfaces. In particular, it is true that all of the geodesics that stay in the smooth part of the surface are closed, and their ...


8

No, for any knotted curve. Such a curve would bound an embedded disk on either side, and, therefore, would be unknotted.


8

Let me give a standard example of a closed incomplete manifold with flat affine structure, whose 2-dimensional version is essentially the example from the comment of Misha. Consider $R^n\setminus 0$ and the action of the group $(Z, +)$ generated by the homothety $$ p\mapsto 2 p.$$ The action preserves the flat affine connection, and also (if the dimension ...


8

This is not always possible, consider for example the punctured (at $(0,0)$) plane with the usual flat metric, and the points $(-1, 0)$ and $(1, 0).$


8

Herbert Busemann defined a geodesic as "a locally isometric map of the real axis" in a metric space. So on his point of view, a geodesic is not primarily length-minimizing but length-preserving. In his 1955 book, The Geometry of Geodesics, he used this definition (p. 32), and a few other properties to prove that: (7.9) A geodesic exists through any two ...


7

If the k-flats are compact, then the space must be symmetric (Molina-Olmos, J. Differential geometry 45 (1997) 575-592; see also Proc. Amer. Math. Soc. 129 (2001), 3701-3709). Homogeneous spaces (non-symmetric and irreducible) with the property that every geodesic is contained in a k-flat ($k\geq 2$) can be constructed as follows (due to Ernst Heintze): ...


7

This is a length-metric, that is any two points $x$ and $y$ can be joined by a path with length arbitrary close to the distance from $x$ to $y$. Further, your metric space is locally bi-Lipschitz to the Euclidean space, in particular it is locally compact. If your space is complete then by Hopf–Rinow theorem it is geodesic. For sure your space is locally ...


7

Burns and Donnay proved that every surface (including a sphere) admits a Riemannian metric that makes the geodesic flow ergodic with respect to Liouville measure, and hence topologically transitive (there is some $v\in SM$ whose orbit under the geodesic flow is dense in $SM$, in other words, the corresponding geodesic is dense in $SM$ in the sense you ...


7

Here is a geometric way that turns out to be equivalent to Robert's answer (i.e., to the Klein-Grifone-Foulon approach to connections associated to a second order ordinary differential equation of a manifold). Let $\phi_t : TM\setminus 0 \rightarrow TM\setminus 0$ be the (local) flow of the second order equation, and let $D\phi_t : T(TM\setminus 0) \...


7

There is a classical example by Hartman that shows failure of uniqueness for $C^{1,\alpha}$ metrics. (P. Hartman, On the local uniqueness of geodesics, Amer. J. Math. 1950). You could lower the regularity if the metric is smooth off some hypersurface but globally only $C^{0,1}$, see for example our recent review: On geodesics in low regularity Moreover, ...


7

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-defined for all $v\in T_pM$ of $g$-length less than $\epsilon$), then, on $B^g(p,\epsilon)$ the squared $g$-distance from $p$ is a smooth function $\rho:B^g(p,\...


7

Yes, such examples do live. The following answer was given in "Metric spaces of non-positive curvature" by Bridson and Haefliger; thanks to GGT and Moishe Kohan.


7

I suggest that you look at what happens in the universal cover given by the Poincare disk model. The five geodesics are quite easy to see. In the picture below, the greyed zone is a fundamental domain for the pair of pants. The five sought geodesics are in bold. The book of Peter Buser, geometry and spectra of compact Riemann surfaces, is a good reference ...


6

I assume that imranal asks how to find numerically a geodesic connecting two given points if the connection is given. One way to do it is to implement the solution of the ODE system he wrote in his question numerically (there are many effective ways for it) and then use this implimintation to find the correct initial velocity vector $(u', v')$ ...


6

Let $k$ denote the value of the constant sectional curvature of $M^m$ and let $X_k$ denote the unique simply-connected complete $m$-dimensional manifold of the constant curvature $k$. Then, since $M$ is simply-connected, there exists a map $dev: M^m\to X_k$, called a developing map of the metric on $M$, which is a local isometry (it is not, in general 1-1). ...


6

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 the symmetric space, which is the dimension $r$ of a maximal abelian subspace $\frak{a}$ in $\frak{k}^\perp\subset \frak{g}$, where $\frak{k}\subset\frak{g}$ are ...


6

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'(...


6

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 (1976), 85–119.) proved (roughly speaking) is that the set of Zoll metrics conformal to the round metric that are sufficiently near the round metric (in an ...


6

There are integral homology spheres in the following Thurston geometries: $S^3$, $\mathrm{PSL}(2,\mathbb{R})$, and $H^3$. No manifold of the latter two types (when equipped with any metric) can have all of its geodesics being closed. This follows from observing that the fundamental group contains a non-trivial free group, and then a Gromov-Hausdorff ...


6

This is a comment, not an answer, but it's too long to fit into a comment window. I don't know of a 'generic' condition on metrics on the torus that would guarantee that there are no null-homotopic closed geodesics, but then I'm not sure what 'generic' is supposed to mean. For example, would it suffice to find some sort of 'open' condition (for example, ...


6

No such bound $C(\gamma)$ exists, even when $S$ is the two-sphere and even assuming that all geodesics considered are simple. Here is the example (which generalises to surfaces with genus). Suppose that $S$ is the two-sphere. Pick four open disks $(D_i)_{i = 1}^4$ whose closures are closed disks and which are pairwise disjoint. (For example, use small ...


6

Yes. Consider segments of $u$-coordinate curves between two $v$-coordinate curves $\{u=u_1\}$ and $\{u=u_2\}$. These segments are a family of geodesics of the same length $|u_1-u_2|$. By the first variation formula, this implies that each of these segments meets the bounding lines $\{u=u_1\}$ and $\{u=u_2\}$ at equal angles. Since $u_1$ and $u_2$ are in fact ...


5

No = there are no non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex. Note that any non compact $n$-manifold $M$ contains a closed subset homeomorphic to $[0,\infty)\times D^{n-1}$ Indeed choose a bounded metric on the manifold $M$ and a minimizing geodesic $\gamma\colon[0,\ell)\to M$ and then pass to ...


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