# Questions tagged [geodesics]

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### Geodesic laminations on the 4-punctured sphere

Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...
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### Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
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### Compute distance between geodesics and perturbed geodesics on a Riemannian manifold via Jacobi field $\vert J \vert$

I would like to pose a question regarding the distance between a geodesic $\gamma(t)$ and a perturbed geodesic $\gamma_{\epsilon}(t)$ on a Riemannian manifold. Specifically, is the distance controlled ...
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### Geodesics on orthogonal matrix

Let $O(n)$ be the manifold of orthornormal matrix, i.e. $$O(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=I\}.$$ Then $O(n)$ is a submanifold of $\mathbb{R}^{n\times n}$. On $O(n)$, there is a ...
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### A Question about an article by Birman, Series

Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
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### Sweeping out the disk: what comes out?

In 2008, Larry Guth gave a new proof of a theorem of Gromov about the min-max widths of the unit $n$-ball. This states that the $p$-parameter width $\omega_p(k,n)$ (of sweepouts with $k$-dimensional ...
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### Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
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### What integral formula is being used here?

I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some ...
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### Intersections of geodesics in an "almost flat" plane

Let $g$ be a complete metric on $\mathbb{R}^2$, such that: Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes. The integral of the Gaussian curvature in $K$ is ...
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### Are geodesics necessarily embedded?

I would like to ask a very basic/naive question. Given a Riemannian or pseudo-Riemannian manidold equipped with the Levi-Civita connection, is it known that all solutions of the geodesics equation are ...
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Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ... • 1,349 9 votes 1 answer 336 views ### Do geodesics avoid regions where the curvature diverges? Let$(M^2,g)$be a Riemannian manifold, with manifold boundary$\partial M$. We assume that the metric degenerates at the boundary, in the sense that the (Gauss) curvature diverges like$K \to +\infty$... • 4,912 1 vote 0 answers 74 views ### Translate of a geodesic that goes through a fixed point on$\mathbb{H}$Consider the complex upper half plane$\mathbb{H}$with the hyperbolic geometry. Fix a point$z \in \mathbb{H}$and also a geodesic$c$. I want to find a hyperbolic translation$\gamma c$passes that ... • 577 6 votes 1 answer 116 views ### On properties of Besse spheres Let$(\mathbb{S}^2,g)$be a Besse sphere, that is, a Riemannian sphere all of whose geodesics are closed. By a result of Gromoll and Grove, all the geodesics are simple (no self-intersections) and ... • 2,075 1 vote 0 answers 59 views ### Showing bound$\|\nabla_t \dot{\tilde{x}}_Y(h, 0)\| \le L \|Y\|_{2, \infty}$for smooth homotopies of geodesics This question pertains to Lemma 3.5 of this article. Let$M$be a smooth Riemannian manifold and$\gamma$some geodesic with respect to the Levi-Civita connection$\nabla$. For any$C^2$vector field$...
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Is there a construction in Riemannian geometry which relates the gradient flow of a function on a manifold with a certain metric with geodesics on another related manifold with its own metric?
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### A question on convexity and conjugate points

Let $(M,g)$ be a compact smooth simply connected Riemannian manifold with a smooth boundary. Assume also that $(M,g)$ does not have any conjugate pairs of points. Let $\Gamma \subset \partial M$ be a ...
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### If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?

This is a cross-post. Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric. Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...
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### Sufficient condition for geodesic convexity/connectedness

Let $(\Sigma,g)$ be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the ...
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### Can the existence of geodesics be deduced from properties of the Laplacian?

As I understand it, the semiclassical trace formula in particular relates lengths of geodesics to eigenvalues of the Laplacian. Is it possible to prove that every compact Riemannian manifold has a ...
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### Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths

While going through this paper by Witten and seeing a discussion about different aspects of Raychaudhari Equation and Einstein Field Equation. I want to ask if Raychaudhari Equation find any ...
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### Hyperbolic length of curve that does not enter a collar

Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...
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### Uniform divergence of geodesics in RAAGs

$\DeclareMathOperator\div{div}$Let $(X,d)$ be a metric space and let $\gamma$ be a geodesic in $X$. Roughly speaking, the divergence of $\gamma$ at a point $x\in \gamma$ is a function \$\div:\mathbb{R}...
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