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2 votes
0 answers
97 views

Property of parallel translation in Green and Wu, "On the subharmonicity and plurisubharmonicity of geodesically convex functions"

In the mentionned paper, I am having difficulties in understanding the proof of lemma 2. Roughly, this lemma says that given any separation $\eta$ for the $C^\infty$ topology of smooth paths from $[-1,...
5 votes
0 answers
149 views

Are all linear vector fields geodesible vector fields?

I had already asked this question in MSE then I ask here at MO. Assume that $A\in M_n(\mathbb{R})$ is a non singular matrix. Is the flow of the linear vector field $X'=AX$ a geodesible flow on $\...
3 votes
2 answers
236 views

Lengths of closed geodesics and geodesic segments

Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties: existence of closed geodesics of arbitrarily long length on $M$...
4 votes
0 answers
124 views

Regularity of exponential map for $C^{2,\alpha}$ Riemannian metrics

Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies? Since $\mathrm{exp}_p$ is defined by the ...
2 votes
0 answers
411 views

Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates

Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle $$TM \vert_{\...
2 votes
1 answer
85 views

Jacobi fields in singular metric on quotient space

Consider the square $\Omega = (0,\pi) \times (0,\pi/2) \ni (r,\theta)$ endowed with the Riemannian metric \begin{equation} f^2 \big(\mathrm{d} r^2 + \sin^2(r) \, \mathrm{d} \theta^2 \big), \end{...
0 votes
1 answer
162 views

Going from piecewise to genuine geodesic without decreasing number of intersections?

Let $(M^2,g)$ be a complete, two-dimensional Riemannian manifold be given; also given is $\gamma: [0,\infty) \to M$, an injective geodesic in $M$. Suppose there are two geodesic segments $\gamma_i : [...
0 votes
1 answer
236 views

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 ...
9 votes
1 answer
344 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 $K \to +\infty$...
3 votes
0 answers
210 views

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 ...
5 votes
0 answers
85 views

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 ...
-1 votes
2 answers
298 views

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 ...
2 votes
1 answer
303 views

2 questions about loops and negative curvature

$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$ 1) will the geodesic in the free homotopy ...
5 votes
2 answers
732 views

Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

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 ...
8 votes
1 answer
330 views

trapped geodesics

Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary. We call a point $p \in M$ regular if there exists a geodesic of finite arc length passing through $p$ with end points on $\...
7 votes
2 answers
434 views

Convexity in co-ordinate charts of geodesic balls

Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$. I would guess the following is true: for $\epsilon$ sufficiently small, the $g$-geodesic ...
1 vote
1 answer
164 views

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 ...
3 votes
0 answers
109 views

Application of Santalo’s formula

Suppose that $(M,g)$ is a compact smooth Riemannian manifold with a smooth boundary and suppose that $f$ is a smooth function on $M$ with the property that $$ \int_I f(\gamma(t))\,dt=0,$$ for any ...
23 votes
2 answers
1k views

Can we make distances in a finite subset of a manifold whatever we want?

Given a connected smooth manifold $M$ of dimension $m>1$, points $p_1,\dots,p_n\in M$ and positive values $\{d_{i,j};1\leq i<j\leq n\}$ satisfying the strict triangle inequalities $d_{i,j}<d_{...
2 votes
0 answers
102 views

Distance and initial velocity of the shortest path along a smooth curve in a manifold

Let $(M,g)$ be a Riemannain manifold and let $p\in M$. Let $\gamma:[0,1] \to M$ be a smooth curve and let $p \notin \gamma([0,1])$. Assume further that for each $t \in [0,1]$ there is a unique (unit ...
4 votes
0 answers
106 views

Geodesic foliations of open manifolds foliated by hyperbolic spaces

It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700). Suppose a complete ...
2 votes
0 answers
149 views

Comparison of sum of vectors and exponential map on a Riemannian manifold

Suppose $M$ is a simply-connected complete Riemannian manifold with bounded sectional curvature $\delta \leq K \leq \Delta < 0$. Let $p_0\in M$. Define the sequence of points $p_1, \ldots, p_n$ by $...
3 votes
2 answers
347 views

Direct calculation of the Fisher distance via Riemannian geodesics

I'm looking for a reference for a direct calculation of the Fisher distance (to avoid overloading the term "metric") $d_F(x,y) := 2 \cos^{-1} \sum_i \sqrt{x_i y_i}$ as the geodesic distance ...
1 vote
0 answers
141 views

Injecitivity radius of Sasaki metric

Suppose we have a compact riemannian manifold $(M,g)$ and we endow $TM$ with the Sasaki metric $\tilde g$. Now I am interested in understanding the injectivity radius of $(TM,\tilde g)$ but I am ...
9 votes
0 answers
515 views

Volume of a geodesic ball in $\operatorname{SL}(n) / {\operatorname{SO}(n)}$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son Question: What is the volume of a ...
2 votes
1 answer
232 views

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 ...
3 votes
3 answers
525 views

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 ...
2 votes
0 answers
276 views

Geodesics and potential function

I try to assemble concepts of differential geometry for my own comprehension of the subject. I understand a manifold is a higher dimensional surface. It has a metric which perform inner product in the ...
6 votes
0 answers
160 views

Almost-geodesics on a Riemannian Hilbert manifold which are still almost geodesics in some submanifold

Let $H$ be a separable infinite dimensional Hilbert space, and consider it as a Hilbert manifold in the usual way (that is, with the single chart with the identity map). It is known that there always ...
0 votes
1 answer
203 views

Riemannian metrics on matrix space for which the restriction of trace function to each complete geodesic is a bounded function

Edit: According to comment by Leo Monsaingeon I revise my question: Is there a Riemannian metric on $M_n(\mathbb{R})$ for which the function $trace$ is a bounded function on every complete(whole)...
4 votes
1 answer
1k views

Geodesic convexity and the Geometric Hessian

This is an elementary question in differential geometry. We know that for a smooth real-valued function $f$ defined on an open geodesically convex set of a Riemannian manifold $ \mathcal{X} \subset \...
12 votes
1 answer
937 views

Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...
1 vote
0 answers
85 views

Conjugate points and Jacobi matrices

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 3$ and let $\gamma:[-2,2]\to M$ be a geodesic that does not contain any conjugate points on $[-2,2]$. I have two questions, as ...
3 votes
0 answers
46 views

Conjugate points for a family of generalized curves

Let $(M,g)$ denote a compact smooth Riemannian manifold with boundary and let $\mathscr F$ denote a family of smooth curves $\gamma$ such that they solve $$ \nabla^g_{\dot \gamma} \dot\gamma = F(\...
1 vote
0 answers
303 views

time-derivative and differential of a geodesic flow

I came across the following question in relation to another question. Let $X\colon M \to TM$ be a vector field over a manifold $M$ and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" for the vector ...
2 votes
0 answers
260 views

A geometric property of certain Lie groups

I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive ...
0 votes
0 answers
562 views

directional derivative along geodesic flow of vector field

A rather elementary question for the differential geometers. Let $M$ be a Riemannian manifold, let $X\colon M \to TM$ be a vector field, and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" of the ...
3 votes
0 answers
192 views

Is there a family of Riemannian manifolds with explicitly solvable geodesics for manifold M?

By this paper https://www.mat.univie.ac.at/~michor/rie-met.pdf for any given manifold $M$, there is a Hilbert manifold (I believe its a Hilbert manifold, if not it is still an infinite dimensional ...
3 votes
0 answers
531 views

Geodesics (Local vs Global)

Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
6 votes
1 answer
577 views

Compactness theorem for minimal surfaces

I am a bit confused about the statement of Theorem 1.1 in this paper by Brian White. For convenience, I will restate it here. Theorem: Let $\Omega$ be an open subset of a Riemannian $3$-manifold. ...
3 votes
0 answers
60 views

Transformation between nearby tangent planes [closed]

This question is kinda long, but the picture is quite clear. Question: Let $(M,g)$ be a Riemannian manifold, $p$ a point on $M$, $U$ an open neighborhood of $0\in T_pM$ such that $exp_p|_U$ is a ...
3 votes
0 answers
97 views

non-self-intersecting geodesics

Suppose $(M,g)$ is a smooth compact orientable Riemannian manifold of dimension $d \geq 3$ with a smooth boundary $\partial M$ and let $\gamma$ be a maximal geodesic in $M$ starting from a point $p \...
9 votes
1 answer
355 views

k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$. Question. Are ...
15 votes
4 answers
918 views

Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...
11 votes
1 answer
529 views

Length decreasing homotopies of curves

Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement: Statement. ...
6 votes
0 answers
355 views

Higher order variations of Riemannian geodesics

Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic. There is a well established theory of the first order ...
15 votes
1 answer
936 views

Existence and uniqueness of geodesics in low regularity

Consider a Riemannian manifold $(M,g)$. How much regularity is required of $g$ so that for any $x\in M$ and $v\in T_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to ...
3 votes
0 answers
476 views

Understanding the domain of the "volume density function" on Riemannian manifold

I have some trouble on understanding the domain of the "volume density function" on Riemannian manifold. Putting the volume density function in quote means actually I am working on the function ...
2 votes
1 answer
1k views

When are geodesics straight lines?

Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients ...
7 votes
1 answer
423 views

3-manifolds with all geodesics closed

A theorem of Bott states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \...