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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Why is the set of singular points of starlike boundary $\Gamma$ closed?
I'm reading Geometric Inequalities of Yu. D. Burago and V. A. Zalgaller. I don't understand why $E_1$ is closed in the proof of the following lemma.
Several definition.
Suppose $ \Omega $ is a ...
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Inversive product of tow ultraparallel geodesics in the hyperbolic plane is $\cosh{\rho}$
This is Lemma 7.17.3 in Beardon:
Lemma 7.17.3 Let $L$ and $L'$ be geodesics in the hyperbolic plane. Then the inversive product $(L,L')$ is $\cosh{\rho(L,L')},1,\cos{\phi}$ according as $L,L'$ are ...
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On the definition of pregeodesics as stationary points of the length functional in (pseudo-)Riemannian geometry
On a Riemannian manifold $(M,g)$, pregeodesics can be defined either of the following two equivalent ways:
curves for which the (Levi-Civita) acceleration $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ is ...
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Does a coercive Riemannian metric imply geodesic completeness?
Let $M \subset \mathbb{R}^n$ be a bounded open (in the euclidean sense) set. Define a smooth function $Q:M \to S_{++}^n$, where $S_{++}^n$ denotes symmetric positive-definite $n \times n$ matrices. ...
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Is parallel transport or a connection needed for geodesic computation?
I have been reading a bit more about differential geometry and I'm interested in it from the practical computational perspective. I have seen some places where it is mentioned that you cannot move ...
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Big picture: Geodesics on a spherical surface embedded in $\Bbb R^3$
Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$
Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in ...
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Geodesics connecting two points in the hyperboloid model of the hyperbolic plane
Let $\mathbb{H}^2$ be the upper sheet of the hyperboloid defined by $x^2+y^2-z^2=-1$ in three-dimensional Minkowski space $(\mathbb{R}^3, g_M)$, where $g_M = \text{diag}(1,1,-1)$. In other words, ...
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Special orthogonal matrices, Geodesic and Manifold
My task is to find some manifold on stoichiometric matrices and find the geodesic distance between these nodes (not the euclidean distance).
Here's my idea so far:
Suppose we are given a matrix N of ...
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Proof that geodesics have constant speed [duplicate]
Let $(M,g)$ be a Riemannian manifold. Let $(\varphi,U)$ be a chart. A curve on that chart $\gamma(t)=\varphi^{-1}(x^1(t),...,x^n(t))$ is a geodesic if it solves the geodesic equation $$\ddot{x}^i+\...
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Quasi-geodesic rays are closed to geodesic rays in proper hyperbolic geodesic spaces
We define the boundary of a hyperbolic metric space $\partial X$ as the equivalence classes of geodesic rays up to finite Hausdorff distance and $\partial_q X$ as the equivalence classes of quasi-...
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Question on a stable geodesic lamination on a closed hyperbolic surface
Let me first state a theorem in Casson-Bleiler Automorphisms of Surfaces after Nielsen and Thurston.
Theorem 5.5: Let $h:F\to F$ be a non-periodic irreducible automorphism of a closed orientable ...
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Proof of length of a tangent vector of a geodesic is constant on Finsler Manifold
In proof of Proposition 2.2 of the book Introduction to Modern Finsler Geometry by Shen and Shen I have faced the following problem:
Proposition 2.2: If $\sigma(t)$ is a geodesic on a Finsler manifold ...
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Is a stationary point for length functional automatically a local minimum?
Given a Lagrangian $L:TM \rightarrow \mathbb{R}$ defined on a tangent bundle, Hamilton's principle states that a curve $\gamma:[a,b] \rightarrow M$ is a stationary point of the action functional $S[\...
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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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Proof of the uniqueness of geodesics.
In my notes, there is a proof of the fact that if $c_1,c_2:I \to M$ are geodesics from an interval $I \subset \mathbb{R}$ to a smooth manifold $M$, presumably semi-riemannian, where $c_1(a) = c_2(a)$ ...
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Geodesics without calculus of variations
Can you compute geodesics by treating it as a problem where you want to minimize the length of a curve through two points on a specified surface while having the constraint that the curve must reside ...
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Examples and characterizations of totally geodesic maps
A totally geodesic map between two Riemannian manifolds is a map that carries geodesics of the domain manifold to geodesics of the target/co-domain manifold, i.e. following the definition from the ...
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Totally geodesic diffeomorphism and isometry.
Suppose $(M,g_M),(N,g_N)$ are two Riemann manifolds of the same dimension, and $f$ is a totally geodesic diffeomorphism between them, is it true that $M,N$ must be isometric (probably not through $f$, ...
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Properties of exponential map of convex sum of metrics
Suppose we have a Riemannian manifold $M$ with two complete metrics $g_0$, $g_1$ such that their curvatures are bounded and injectivity radii are bounded away from zero.
My goal is to find a uniform ...
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Geodesics in the Hyperboloid Model
Let $\mathbb{H}^2$ be the upper sheet of the hyperboloid defined by $x^2+y^2-z^2=-1$ in three-dimensional Minkowski space $(\mathbb{R}^3, g_M)$, where $g_M = \text{diag}(1,1,-1)$. In other words, ...
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Implications of requiring that a family of curves in a Lie group be geodesics of a Riemannian metric
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and identity element $e$. I want to define a Riemannian metric on $G$ such that the curves $\gamma_{X, g}(t) = \exp(tX)\cdot g$ with $t\in \...
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Existence of geodesically convex neighborhoods in semi-Riemannian manifolds.
I am studying the text by Barrett O’Neill referenced below. On page 130, O'Neill states, as Proposition 7, that every point in a semi-Riemannian Manifold has a convex neighborhood. Convex is defined ...
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Reparametrization of geodesics (especially in the context of geodesic flow)
The following is in the context of Riemannian geometry, but the pseudo-Riemannian case is just as important to me (not directly addressed here). I want to understand what freedom we have to ...
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Implications of requiring that one-parameter subgroups of a Lie group be geodesics of a Riemannian metric
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and identity element $e$. I want to define a Riemannian metric on $G$ such that the one-parameter subgroups $\gamma_X(t) = \exp(tX)$ with $t\in\...
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Area of a triangle on a torus
On a sphere the area of a trinagle is $A=\dfrac{\pi r^2}{180^{\circ}}\cdot(\alpha+\beta+\gamma-180^{\circ})$ where $\alpha, \beta, \gamma$ are the angles of the triangle.
Is there a similar easy ...
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Help Understanding Perelman's $\mathcal{L}$-geodesic Equation
I'm reading a paper on Perelman's solution to the Poincare conjecture. The paper derives a geodesic equation by minimizing the following $\mathcal{L}$-length of each curve $\gamma(\tau)$, $0<\tau_1 ...
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Can Jacobi Fields be used to explicitly calculate Lyapunov exponents for geodesic flows?
My background is in geometry, and I have just become interested in the ergodic theory of geodesic flow. In Sarnak's 1981 paper
"Entropy estimates for geodesic flows," the following ...
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Ray Tracing In Mathematical Spaces
I really enjoyed the Not Knot video, but I don't fully understand the mathematics which is going on there.
They are animating the space of the complement of the Borromean rings, but you can't just ...
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Marked points on fibers of $TS^2$ can trace out a helix during (parallel) transport.
It is claimed here that a mark on a rod parallelly transported by an observer moving along a geodesic $\gamma(t)$ on a smooth manifold $M$ (with a metric $g$) will trace out a helix, instead of a ...
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Prove that if the cone is cut along a generator and flattened into a plane, then the geodesic becomes a straight line.
The is a problem from Differential Equations with Applications and Historical Notes by George F. Simmons. The chapter is about the calculus of variation.
The hint is to represent equation of a right ...
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What type of curve follows a surface such that the line is straight when the surface is flattened
Given an arbitrary surface, a point on that surface, and a direction, what is the name of the type of curve which lies in the surface and which twists and turns along with the curvature of the surface,...
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Preconditioned gradient flow keeps boundedness
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a real analytic function with $\inf_{x\in\mathbb{R}^n} f(x) > -\infty$. If we know the solution $x:[0,\infty)\rightarrow\mathbb{R}^n$ to the vanilla ...
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Find the equation of a curve between two points on a cone respectively on a cylinder and calculate a new point on this curve
I have a curve between two points $P_1 (x_1, y_1,z_1)$ and $P_2 (x_2, y_2, z_2)$ which both lie either on the surface of a cylinder (case $A$) or on the surface of a cone (case $B$). The curve is the ...
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If the coordinate curves are geodesics, then $\partial_v E=\partial_u G=0$
We have a parametrized regular surface and its coordinate curves in some parametrization are geodesics. I started by noting that, if $g_u, g_v$ denote the tangent vectors to those curves, then $\...
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How does a constant scaling of the metric affect the geodesics? [duplicate]
Let $(M,g)$ be a Riemannian manifold, and let $\lambda$ be a positive number. Then we know $(M, \lambda g)$ is also a Riemannian manifold.
Also, if we have a point $p$ in $M$ and a vector $V$ in $T_pM$...
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Understanding the $k$-dimensional totally-geodesic submanifolds in the Poincare Ball model of the Real Hyperbolic space
I was looking at the submanifolds in the Poincare Ball model of the real hyperbolic space. It has been mentioned in various places that the totally geodesic hypersurfaces are the parts of the sphere ...
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Solve $\inf_{\mu_0=\mu,\mu_1=\nu}\{\int_0^1\|v_t\|_{L^2(\mathbb{R}^d)}|\partial_t\mu_t+\text{div}(\mu_tv_t)=0\}.$
Let $\mathcal{P}_{2,ac}(\mathbb{R}^d)$ be the space of all probability measures on $\mathbb{R}^d$ with finite 2nd momentum and absoulute continuous w.r.t. the Lesbegue measure. It is known that if we ...
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Geodesics in $\mathbb{H}/\mathrm{SL}_2(\mathbb{Z})$
I am trying to solve an Exercise in the book Audrey Terras - Harmonic Analysis on Symmetric Spaces and Applications I-Springer (1985) and I struggle to find good justifications.
In the exercise, we ...
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Simple closed geodesics meeting at only one point
I am trying to prove the following corollary from the book Differential Geometry of Curves and Surfaces by Kristopher Tapp, p. 332.
If $S$ is a regular surface that is diffeomorphic to a cylinder and ...
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There is no metric that makes the integral curves of the field $X=(y,-x)$ geodesics?
I used the fact that $\nabla_X X=0$, which implies that the integral curves of $X$ are geodesics, to find the Christoffel symbols(the non-zero ones are $\Gamma_{22}^1=1/x,\Gamma_{11}^2=1/y$). I then ...
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Is there a method to find the length of a geodesic length from its parameterisation and 'velocity'?
I am attempting to implement this, pages 11/12 method to plot geodesic equations on the surface of an object, beginning with the sphere. I would like to be able to guarantee that the geodetic length ...
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Pointwise convergence of geodesics implies uniform convergence
Show that if $\gamma_{n} \colon [0, 1] \rightarrow M$ is a sequence of geodesics in a Riemannian manifold $M$ such that $\gamma(t) = \lim_{n \rightarrow \infty} \gamma_{n}$, then
$\gamma$ is a ...
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Can two geodesics meet tangentially on a Finsler manifold?
On a Riemannian manifold, it is well-known that two distinct non-closed geodesics can not meet tangentially, that is, if they meet at time $t_0$ and $t_1$, then they will meet at different angles. The ...
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Question about geodesic
On the Wikipedia page for Geodesic, it's stated that a curve $\gamma : I → M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v \geq 0$ such that for ...
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Uniformly converging geodesic sequence yields a geodesic
while solving a problem John Lee's Introduction to Riemannian manifold exercise 6-17 (exercise itself is not important in this question, however it was the question asking to prove the existence of ...
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A question about an article by Birman, Series
I've asked this question on mathoverflow, but I thought it may be helpful to restate here since I guess it might be a not so hard question to solve.
The link
Birman and Series in their article ...
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How to calculate a co-ordinate transformation between two metrics at a moving boundary?
While following a derivation for asymptotic solutions to a scalar field in Schwarzschild spacetime, it became necessary to approximate the interior null co-ordinates as a function of the exterior null ...
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How to find center of an arc given a geographical start point, end point and arc angle?
I am working in a way of drawing transitions in some kind of "Flight Plan" between two "waypoints" using the heading at the beginning of the arc and at the end of an arc. Given an ...
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Is a curve in a 1-dimensional manifold always a geodesic for some parameterization?
Consider a curve embedded in a 1-dimensional manifold. Since the manifold itself is one-dimensional, there are no alternative curves that can be taken. Based on this observation, is it correct to say ...
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Derivation of Geodesic Equations for Surfaces Embedded in R3
On page 272 of Banchoff and Lovett's book "Differential Geometry of Curves and Surfaces," they make the following argument (paraphrasing):
For a parameterization of a surface embedded in $\...
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