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Questions tagged [geodesic]

A geodesic is a generalisation of the notion of a straight line to curved spaces, and can be sometimes be thought of as the locally shortest or extremal path between points.

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Proof a curve is a geodesic on a sphere S

Here's my problem on the textbook: Prove that the curve $c(t) = (6\cos t, 6\sin t, 0) $ is a geodesic on the sphere $S$: $x^2 + y^2 + z^2 = 36$ The definition of a geodesic in my text book is: "A ...
Kiệt Lê Nhân's user avatar
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In a geodesic triangle, is the longest side opposite to the largest angle?

If I have a complete (smooth) Riemannian manifold $(M,g)$ and three points on it, that I connect with distance minimizing geodesics, will the longest edge be opposite to the largest angle? In ...
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Is squared geodesic distance about a point convex when defined on a convex ball centered at that point?

Let $\mathcal{M}$ be a complete Riemannian manifold. Let $x \in \mathcal{M}$, and let $r_x >0$ be the convexity radius at $x$. Let $B \subset \mathcal{M}$ be a geodesic ball centered at $x$ with ...
Spencer Kraisler's user avatar
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Details in the proof of a local version of Cartan-Ambrose-Hicks theorem

I am very sorry for the long length, but most of it is notions conventions and there isn't much argument and deduction. Let $ M^{n} $ and $ \bar{M}^{n} $ be two Riemannian manifolds. Fix base points $...
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Why can a vector field defined only on a curve be evaluated its covariant derivative along the non-tangent direction of the curve?

I am reading the Riemannian geometry part of complex differential geometry by Fangyang Zheng. I have a problem when reading the definition of index form: Let us discuss the index lemma and the second ...
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Is singular disc diagram same as Van Kampen diagram ? How to draw singular disc diagram for hyperbolic groups?

I couldn't find an elaborate description to draw singular disc diagram for hyperbolic groups (or word hyperbolic groups) and some of the google search result is showing that it similar as Van Kampen ...
mrinal nath's user avatar
3 votes
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An "almost" geodesic dome

A regular $ n$-gon is inscribed in the unit circle centered in $0$. We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
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Geodesics in non-convex subsets of $\mathbb{R}^2$.

Let $U \subset \mathbb{R}^2$ be a non-convex connected subset. Let $x,y \in \text{int}(U)$ be such that the straight line joining them does not lie entirely in $U$. Let $\gamma$ be the shortest path ...
Ajay Kumar Nair's user avatar
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The geodesic exponential of a sum

Let $(M,g)$ be a Riemannian manifold, $x\in M$ and $u,v\in T_xM$. I'm wondering what is known about $\exp_x(u+v)$. One obvious guess would be: let $V$ be the unique $\nabla^g$-parallel (here $\nabla^g$...
Jaime Pedregal's user avatar
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Radial geodesic passing through center

In my class we briefly treated the following proposition: Given a geodesic ball $B=exp_{p}(\bar{B})$ and $p,q\in B$, there exists a unique geodesic $\gamma:[0,1] \to B$ such that $\gamma(0)=p$ and $\...
Alessandro Vagni's user avatar
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Doubts about geodesics equation derivation from Lagrange's equations

I want to derive the geodesic equation from a Lagrangian pov, so I consider a Lagrangian $L(q,\dot{q})$ given only by the kinetic energy wrote as the quadratic form of the kinetic matrix $A(q)$, i.e (...
a very confused mathematician's user avatar
3 votes
1 answer
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Closed geodesics of a non positively curved manifold are minimizing

As the title says, closed geodesics of a complete non positive sectional curvature manifold should be minimal in their free homotopy class. This should be well known but I don't know a reference. I ...
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Petersen’s proof of the Cartan-Hadamard Theorem

I was studying the proof of the Cartan-Hadamard Theorem, and the various lemmata that lead to it. Since I couldn’t understand my professor’s proof (and he also acknowledged it was flawed), I ...
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Hyperbolic parallel lines of constant distance

Is it possible to have two distinct hyperbolic lines at a constant distance from each other? To clarify: I mean that two lines $p$ and $q$ are at a constant distance from each other if $\exists c\in [...
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Parameter curves are geodesic then $E_v=G_u = 0$

I'm trying to solve question $\text{4/II/12G}$ of this pdf: A special case $F=0$ of this question is proved in another post For $(\text{i})$: The tangent vector to the unit-speed curve $u=c$ is $\...
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Geodesic tangent space is a vector space?

I have very little knowledge of Differential Geometry and I'm stuck while reading about General Relativity. Consider defining something called a null geodesic tangent space, in analogy with the ...
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Equivalent condition for geodesic in $\mathrm{SO}(m)$

Consider the special orthogonal group $\mathrm{SO}(m)$ equipped with the standard left-invariant Riemannian metric $$g(X,Y)=\operatorname{tr}(X^T Y),$$ where we identify $\mathrm{T}_e\mathrm{SO}(m)$ ...
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surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics.

I try to solve Exercise $122$ on page 40 of this pdf Show that the surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics. ...
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Angle between initial vector and vector transported along a polygon in the hyperbolic plane

Consider the Poincare half-plane model of hyperbolic geometry. Consider the polygon, bounded by the following 4 curves: $x=-2a, (x+2a)^2+y^2=36a^2,(x+a)^2+y^2=a^2,(x-2a)^2+y^2=4a^2$. By what angle ...
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1 vote
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Questions about a differential geometry exercise

The exercise comes from an old exam from the 90s at my university Let $\varphi(u,v) = (v^2-u,u,u-v) \quad u,v \in \mathbb{R^2} \quad S=im(\varphi)$ a) Prove that $S$ is a regular surface and that a ...
jackes gamero's user avatar
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Does there exist a harmonic map $F: (S,g_0)\to T^2 $ for which $F(\alpha_u(t))$ and $F(\beta_v(t))$ are all geodesics?

Let $S=[0,1]^2$. Ignoring issues having to do with boundaries and corners, a chart is a diffeomorphism $\varphi \colon S \to S$. Let $g_0$ denote the flat (euclidean) metric. Given a chart $\varphi \...
zeta space's user avatar
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If $\gamma_v$ is geodesic, then $\gamma_{sv}$ is geodesic

I'm trying to solve the following: Let $p\in S$ ($S$ is a surface) and $v\in T_{p}S$ and a $\delta > 0$ s.t. $\gamma_{v}:\left(-\delta,\delta\right)\to S$ is a geodesic and $\gamma_{v}\left(0\...
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Are the geodesics given by the semicircles in the Poincaré upper-half plane parameterized by arc length?

Given a parameterized surface such that $E = \frac{1}{v^2}$, $F = 0$, $G = \frac{1}{v^2}$ I am asked to prove for curves of the form $(u(s),v(s)) = (a + r\tanh(s),r\operatorname{sech}(s))$ that: (1) ...
jackes gamero's user avatar
5 votes
1 answer
121 views

Geometry of origami saddle surfaces made of five or six square paper sheets connected around a point

I connected five and six square paper sheets (which are all initially flat and have the same dimensions) using tapes to create two smooth saddle surfaces (see below), but I couldn't figure out the ...
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Using Geocentric or Geodedic latitude with Great Circle Distance

I've recently started working on a system does some work using geodetic coordinate systems. The data it uses is WGS84 and some of the modern components use a Vincenty Algorithm for distance but there ...
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Hyperbolic surfaces with only one short geodesic

$\textbf{Question}$: Let $R>0$. Does there exist a compact hyperbolic surface $S$ which has one and $\underline{only\ one}$ primitive geodesic of length $\le R$? I am aware of the fact that the ...
Lille Nordmann's user avatar
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A property of the exponential map on flat torus

Let $\mathbb{T}^n$ be the flat torus defined as $\{ z \in \mathbb{C}^n \colon z = (e^{i\theta_1}, \ldots, e^{i \theta_n}), \ \theta \in \mathbb{R}^n \}$ endowed with its standard Riemannian metrics. ...
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Different definitions of the cut locus

Let $(\mathcal{M},g)$ be a Riemannian manifold which is complete and connected. For any $(p,v) \in \mathcal{TM}$, let $\gamma_v$ denote the maximal geodesic starting at $p$ in direction $v$. The cut ...
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Interpolating points on a sphere between two points

I managed to solve it using the following function: given a cartesian point A and point B. the geodesic path on a sphere is defined as: r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1] then normalize r(t)/||...
Jenia Golbstein's user avatar
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Interpretation of geodesic departing angle

I'm trying to calculate the departing angle of a geodesic using several different methods in two different coordinate spaces. Currently I'm testing the following geodesic terminal points: Start (80°W,...
Reinderien's user avatar
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1 answer
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If the metric does not depend on a coordinate $i$, then along a geodesic, $g_{i\mu}\overset{.}{x}^\mu$ is conserved

In the title, $\overset{.}{x}^\mu$ is meant to mean the derivative of the geodesic with respect to the curve parameter, that is, the tangent to the geodesic. This is another of my general relativity ...
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Eikonal equation and Geodesic flow

I am reading some context from computer graphics about using eikonal equation to compute geodesics on triangular mesh, I find some reference e.g text The authors state that if $\| \text{grad} \: \...
ToastaFish's user avatar
3 votes
0 answers
109 views

Strongly convex sets are totally normal

Let $(\mathcal{M},g)$ be a smooth Riemannian manifold. We say that an (open) subset $U \subset \mathcal{M}$ is strongly convex if and only if every pair $p$, $q \in U$ has a unique geodesic of minimum ...
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2 votes
1 answer
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Geodesic Wasserstein space => the base space is also geodesic?

Let $(Z,d)$ be a Polish space, and for $p\geq 1$, consider a metric space $(W_p,d_{W^p})$ defined by The Wasserstein Space $\begin{align}W_p = \{\mu|\mu\textrm{ is a Borel probability measure on Z ...
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Notation clarification: What is the meaning of $\frac{\omega}{\mathrm{d}t}$?

Here is a homework question: Let $M$ be a $C^\infty$ differentiable manifold with the affine connection $\nabla$, $\{e_i\}$ be a local frame, $\{\omega^i\}$ be its dual frame, $\{\omega_{j}^i\}$ be ...
Zoudelong's user avatar
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0 answers
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Does the Tangent theorem for circumferences hold in the sphere $S^2$? [closed]

I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere. The situation that I have in mind is the following. Take two geodesics $\gamma_1$ and $\gamma_2$ ...
Cris's user avatar
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1 answer
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Isometric copy of the quotient $N$ embedded in the domain $M$ when $\pi:M\to N$ is a surjective Riemannian submersion?

Let $\pi:(M,g)\to (N,h)$ be a surjective Riemannian submersion, i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that . $T_pM=H_pM \oplus V_pM$ ( $g_p$-orthogonal ...
Learning Math's user avatar
1 vote
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Unable to visualize geodesic as an integral curve of a vector field in the tangent bundle.

I have began to study Riemannian Geometry and there I encountered a statement that I am unable to feel/understand/visualize. It is the following statement that is still bothering me: Any geodesic for ...
Kishalay Sarkar's user avatar
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Definition of geodesic in metric spaces

My question is closely related to this: On the definition of a geodesic in a metric space I don't understand why in the definition of the geodesic there is the requirement of constant speed. As far as ...
Markus's user avatar
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Are sufficiently small closed geodesic balls compact?

Let $\mathcal{M}$ be a (not necessarily complete) Riemannian manifold. As a result, we cannot claim that closed and bounded subsets are compact. Pick $x \in \mathcal{M}$. Let $r = \text{inj}_{\mathcal{...
Spencer Kraisler's user avatar
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What information does the sign of the index form gives me about the geodesic?

For a Riemannian manifold $(M,g)$ Given a geodesic segment $\gamma:[a,b]\to M$ we define a symmetric bilinear form $I$ , called the index form of , on the space of normal vector fields along $\gamma$...
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Lagrange eq is the geodesic equation if and only if the curve is parametrized by constant speed $∥\dot\gamma (t)∥$

Let $(M, g)$ be a Riemannian manifold. The length of an admissible curve $\gamma : [a, b] \to M$ is defined by $L(γ) = \int_a^b ∥\dot \gamma (t)∥dt$. 1)Compute the Euler–Langrange equations for the ...
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The existence of a larger compact set containing all minimising geodesics of a compact subset

On a hyperbolic surface, is a compact set $K$ necessarily contained in a larger compact set $K’$ so that any two points of $K$ can be joined by a minimising geodesic within $K’$? The question came out ...
Helen's user avatar
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How is the distance between a point and a polygon boundary computed in a spherical or ellipsoidal coordinate system?

I had previously asked this question on https://gis.stackexchange.com/, but it seems like they don't usually handle "pure math" questions, and it was poorly received, so I closed it. PostGIS ...
shadowtalker's user avatar
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Second derivative of function evaluated along a curve on a Riemannian manifold

Let $\mathcal M$ be a Riemannian manifold and $f : \mathcal M \to \mathbb R$ be smooth. Take a smooth curve $\boldsymbol \gamma : I \to \mathcal M$, where $I$ is an open interval of the real line. I'm ...
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Understand the geodesic, $\gamma$, induced by an abelian form $\omega$ and their integral

in my phd thesis, informaly I use this result: Let $(A_i,\omega_i)$, $i=1,2$, where $A_i$ is an annulus and $\omega_i$ is an abelian differential form with a translation structure on $A_i$ that ...
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0 answers
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Computing a curve on a surface which has constant geodesic curvature

The paper Computation of Shortest Paths on Free-Form Parametric Surfaces shows how to compute a geodesic curve on a surface. In this paper they start with two equations which are true for a geodesic ...
adam.lofts's user avatar
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Distance formula in $S^2 \times S^2$ [duplicate]

I am absolutely not a specialist in topology, and this question is quite hard for me... I read about metric tensors, geodesic formulas, but it is difficult for me to understand how to use them in ...
Balfar's user avatar
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Characterization of the Levi-Civita connection through length-extremization property of geodesics

I know that geodesics on a Riemannian manifold equipped with the Levi-Civita connection $\nabla$ have two equivalent characterizations: They are energy-extremizing curves (which is equivalent to ...
Francesco Paronetto's user avatar
1 vote
1 answer
66 views

Is $\vec{0}$ a geodesic? (specifically in $\Bbb{R}^3$ but also in the general case)

In my introductory differential geometry course (in which we only really dealt with $\Bbb{R}^3$ apart from some 'asides' to give context and to give some intuition about generalising some concepts for ...
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