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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Acceleration and Geodesic

The following is from Differential Geometry of Curves and Surfaces. “If a small motorised car is set on a flat surface it will travel forward in a starlight line. What if it is set on the curved ...
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Gauss Map and Geodesic Flow

I was reading chpater (9) of the "Ergodic Theory with a view towards Number Theory" book by Manfred Einsiedler and Thomas Ward. To be more precise, I was trying to understand the connection ...
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Exponential map in the Lie vs. Riemannian category

Let $G$ be a Lie group with a bi-invariant metric. Then the exponential map at the identity in the Riemannian sense, $\operatorname{exp}(v)$, is the same as the exponential map in the Lie sense, $e^v$....
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Nonpositive curvature with closing geodesics.

Let $M$ be a complete Riemannian manifold, $x \in M$ a point, and $\gamma_s(t)$ be a family of geodesics starting at $x$ at time $t=0$, $s \in [-\epsilon,\epsilon]$. The Jacobi field $J(s,t)$ of this ...
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When does there exist a diffeomorphism between geodesics of distinct derivative operators?

Let $M$ be a smooth, connected manifold, and let $\nabla$ and $\nabla'$ be two covariant derivate operators on $M$ such that $\nabla \neq \nabla'$. As we know, a derivative operator $\nabla$ is ...
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Make definition of a geodesic on a smooth manifold independent of parameterization

I’m trying to prove that a regular curve $c$ on a smooth manifold $M$ is a geodesic iff $\nabla_{c’(t)}c’(t)$ and $c’(t)$ are on the same direction for any parameterization $c(t)$. I’m not sure how to ...
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Relation from Douady paper about circle homeomorphisms.

In this paper the authors write down the following relation (p. 5, the last equation before the 3rd section): $$-\frac{1}{2}\ln\left(\frac{1-|z|^2}{|z|^2}\right)=\lim\limits_{r\rightarrow 1}\left[L(z,...
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How to check if other points are on the geodesic between 2 points on a sphere?

Given any 2 points $p_1, p_2 \neq p_1$ on a sphere, how can we check if $p_3$ is on the geodesic from $p_1$ to $p_2$? I think of checking the $||\text{arc } p_1 p_3|| + ||\text{arc } p_3 p_2|| = ||\...
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Derivation of geodesic equations from the definition of parallel vector fieds

I am reading "Differentiable Manifolds: A Theoretical Physics Approach" by Torres del Castillo. On page 116 of the book the author gives the equations for a parallel vector field along a ...
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Stable and Unstable Horocycle Flows of the Unit Tangent Bunble $\mathrm{T^1}\mathbb{H}$ of $\mathbb{H}$

Ergodic Theory with a view towards Number Theory, Chapter 9, Page 287, 288. Let $$\mathbb{H}=\{x+iy \in \mathbb{C}, y>0\}$$ and $\mathrm{T^1}\mathbb{H}$ be the unit tangent bundle of $\mathbb{H}$. ...
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Uniqueness (up to rotation) related to the signed geodesic curvature

Take $S^2$ to be the unit sphere oriented by its outward normal. Let's consider two unit-speed parametrized curves $\alpha=\alpha(s)$ and $\beta=\beta(s)$ in $S^2$. If they have the same signed ...
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Geodesic equation with respect to a Riemannian diagonal metric

I am asked to prove that the differential equations of geodesics on an open set of a pseudoRiemannian manifold $(M, g)$ of dimension $n$ where $g_{ij}=0$ if $i \neq j$ are given by $$ \frac{d}{ds}(g_{...
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geodesics on sphere loring tu help

Loring Tu in his book Differential geometry [page 104] states (in my own words) the following on geodesics on spheres: Consider the 2 sphere of radius $a$ in $\mathbb{R}^3$ . Parameterise a great ...
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Why does a geodesic $\gamma$ have unit speed and $\ddot{\gamma}$ is normal to $S$ at each point

Hi i am reading about geodesics and there in the definition it is mentioned that A curve $\gamma$ on a surface $S$ is called a geodesic if : (a) it has unit speed (b) $\ddot\gamma$ is normal to $S$ ...
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Geodesics of low regularity

Section 3.6 of Kobayashi and Nomizu's "Foundations of differential geometry, volume 1" says "A curve $\tau=x_t$, $a<t<b$, where $-\infty\leq a<b\leq\infty$, of class $C^1$ in a ...
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Geodesic convexity and Exponential map

Given a complete Riemannian manifold $M$, what assumption on $M$ (or just on its curvature) would ensure that for all geodesically convex functions $f:M\to\mathbb R$ and all $x_0\in M$, the map $f\...
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Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions

Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions. Y. Guivarc'h and Y. Le Jan. Page 24. How to make sure that $K$ and $D$ are normal subgroups of the group $G$ to ...
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Highly Recommended References for Hyperbolic Planes and Modular Surfaces

I would be very grateful if one could suggest highly recommended references for Hyperbolic Planes and Modular Surfaces that provide a readable self-contained introduction to these concepts.
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Geodesic equation for the motion of a particle

i know i can write the geodesic equation for a massive particle as: \begin{equation} \dot{x}^{\nu}\nabla_\nu \dot{x}^{\mu}=0 \end{equation} and then we can express this using the 4 momentum, $ p^\mu =...
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On Riemannian geodesics

I am trying to understand Riemannian geometry from the book "Riemannian Manifolds: An Introduction to Curvature" by John Lee. I have the following doubt. A Riemannian geodesic on a ...
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Constant speed parametrization

I am studying the book "Riemannian Manifolds: An Introduction to Curvature" by Lee. The following statement is from that book. Let $S^n$ be the sphere of radius $1.$ The Geodesics of $S^n$ ...
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35 views

Existence of unit speed geodesics

I am reading Lee's book "Riemannian manifolds: An Introduction To Curvature". Let $(M,g)$ be a Riemannian manifold. Fix $p\in M.$ It is well known that the exponential map $\exp_p$ map is a ...
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Prove that there is a closed geodesic in the “inside” of a torus

I want to show that there exists a closed geodesic on an arbitrarily chosen surface where the Gaussian curvature is negative (on the entire surface). One example would be the "inside" of a ...
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Questions about Hopf-Rinow theorem

I am reading about Hopf-Rinow theorem using the Jonh M. Lee "Riemannian manifolds: an introduction to curvature",page 108, Theorem 6.13. My doubts are the next ones: In the book, there is ...
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Non complete connected Riemannian manifold

In John M. Lee "Riemannian manfidolds: an introduction to curvature", page 108, as a previous motivation for Hopf-Rinow theorem, there is an example of non geodesically complete manifold: &...
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$C^{-1/2}$ in hyperbolic distance equation (conformal representation)

Penrose introduces hyperbolic geometry in the second chapter of his book "The Road to Reality". While discussing the conformal representation he introduces a formula to find the hyperbolic ...
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Geodesics on hyperbolic surfaces

Suppose that one has a compact hyperbolic surface realised as a quotient $X=\Gamma\setminus\mathbb{H}$. So in particular, $\Gamma$ consists only of hyperbolic elements of $\mathrm{PSL}(2,\mathbb{R})$ ...
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Parallel translation of a vector on a surface: parallels of the sphere

In V.I. Arnold's Mathematical Methods of Classical Mechanics the first appendix is on Riemannian curvature. He starts with parallel translation of a tangent vector to a surface: Along a geodesic the ...
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$\gamma$ is line of curvature $\iff$ $\frac{d\theta}{ds} + \tau = 0$

I have the following problem in differential geometry. Suppose $S\in\mathbb R^3$ regular surface, and $\gamma:[0,L]\rightarrow S$ regular curve parametrized by arc-length. Suppose that $\theta(s)$ is ...
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Best fitting geodesic on a sphere

Given a list of points (latitudes and longitudes) on a sphere, I want to find the geodesic which minimizes the sum of squared deviations from the geodesic to the points. Is there a nice and efficient ...
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Finding the geodesics on the tangential developable surface of a circular helix.

I am trying to find the geodesics other than the rulings for the developable surface generated by the tangents to a circular helix with $z-$ direction as its axis. The parametrization of one sheet of ...
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Natural parametrization of a geodesic

I want to calculate the geodesic on the surface $f(x,y)=x^2+y^2$ that goes from the origin, with direction $(\cos\theta_0:\sin\theta_0)$, and using a natural parametrization. I used polar coordinates ...
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How to solve the geodesic equations in 2D polar coordinates?

Consider the XY-plane as a surface in 3D-space, parametrized with polar coordinates $u$ an $v$ as $\vec{r}=(v\cos(u),v\sin(u),0)$. For the geodesic equations I get Eq.1: $v~d^2u+2~du~dv=0$ Eq.2: $d^...
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An intuitive argument for proving that existence of interior conjugate points implies non-minimizing

I am learning the book on Riemannian manifold by John Lee. Can someone help me understand the underlined sentence in the text? (I cannot post the image directly due to the rep requirement, please ...
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Geodesics in the symplectization of a contact manifold

As the title suggests, are there any reference that deals with geodesics in the symplectisation $$\Bbb R \times X$$ for a contact manifold $(X,\alpha)$. We equip the symplectisation with the ...
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Differential equation of Geodesics of a Cone

I am working through Tom Kibbles' Classical Mechanics and am struggling to derive an equation in the excercises on calculus of variations. The question is to find the geodesics of a cone with semi-...
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Geodesics on a cylinder: straight line or helix?

It is known that geodesics on a cylinder are helical lines (helix). As a special case of a helix--- straight line. But one can take two points so cleverly that you can draw two helixes between them (...
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Local obstruction to having a complete extension

This question arose while I was reading Helgason's book on symmetric spaces. In chapter IV section 5, one can read the following: Let $M$ be a Riemannian manifold, $p$ a point in $M$. In general it ...
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Geodesic curve induced by the posisitive symmetric matrix

A Riemannian metric $g$ on $M$, n dimensional manifold, is a smooth family of inner products on the tangent spaces of $M$. Namely, $g$ associates to each $p\in M$ a positive definite symmetric ...
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Geodesic Ball on $\mathbb{R}^2$ with Riemannian Metrics that only depends on $r$

I am having a problem that deals with the metric on $\mathbb{R}^2$ parametrized with polar coordinate $g=dr^2+f(r)^2d\theta^2$ for some $f$ such that $f(0)=0,f'(0)>0,f(r)>0$ for $r>0$. It ...
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Geodesic distance of two quaternions

Let $q_0$ and $q_1$ be unit quaternions, and $d(q_0, q_1)$ the angular geodesic distance between the two. I gather that the angular distance can be calculated with the formula: $$d(q_0, q_1) = \lVert \...
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The meaning of geodesic curvature, for a geodesic curve

If $\alpha : [0,1] \to M$ is a geodesic curve in a surface $M\subseteq \mathbb R^3$, then is the "geodesic curvature of $\alpha$" equal to the "curvature of $\alpha$" (defined ...
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How can one calculate the geodesic curvature of arbitrary arcs on a sphere?

Consider that one regular hexagon and one square cut a sphere of radius $r$ as shown in the attached figure. A circular arc of length $s_1$ and a circular arc of length $s_2$ are formed (blue arcs). ...
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How to generalize the Euclidean “unicycle” model?

There is a common system of ODEs known as the unicycle model / Dubin's model which describes the kinematics of an ant-like "unicycle" that can drive forward with some velocity $v(t) \in \...
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If every normal section of a surface is a geodesic, then for every point of the surface, the curvature is the same in any direction.

How should I show that if every normal section of a surface in $\mathbb{R}^3$ is a geodesic, then for every point of the surface, the curvature is the same in any direction? I would like some hints. ...
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A minimizing geodesic that intersect simple closed curve orthogonally

Let $C$ be the trace of a simple closed curve in a regular surface $S$. Let $p\in S-C$. Assume that $p$ is close enough to $C$ that a normal ball about $p$ intersects $C$. Let $q\in C$ be the point of ...
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Defining the exponential map without explicit affine connection

Given a smooth manifold $M$, a smooth vector field $V$, and some $x_0 \in M$, we can define the exponential map through the map $\phi$, where $\phi$ is defined on some open interval of $\mathbb{R}$, $\...
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Geodesic curvature of any circle on a sphere

Assume a sphere of radius $r$. Consider that one hexagon and one square cut the sphere as shown in the attached figure. One circle inscribed within the hexagon with given radius $r_1$ and one circle ...
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Show that this quasi-geodesic ray is not Gromov hyperbolic?

Consider the spiral (t, log(1+t)) (given in polar coordinates); it inherits the Euclidean metric from the plane. I have to show that this spiral (a quasi-geodesic) is not Gromov hyperbolic. In other ...
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geodesic on Riemann manifold and tangent space

Knowing almost nothing about differential geometry, I have to understand this statement from the book that i am reading (page 115): Recall that a geodesic on a Riemannian manifold $M$ is a path $\...

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