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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 ...
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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 ...
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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 (...
3 votes
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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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### 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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### 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 ...
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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) ...
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### 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 ...
3 votes
1 answer
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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 ...
1 vote
1 answer
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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)/||...
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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,...
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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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### 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 ...
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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 $\gamma_2$ ...
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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 ...
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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 ...
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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 ...
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