Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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81 views

Lamé parameters and distance on a curved surface

I was wondering if it is possible to compute the distance between two points which lay on a curved analytical surface. The surface is defined with differential geometry formulae (position vector of ...
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42 views

Orthochronous Lorentz is time preserving and $\operatorname{SL}(2,R)$

Let's consider the pseudosphere in $\mathbb{R}^{1,2}$ given by $x^2+y^2-z^2=-R^2.$ We know that the group $O(1,2)=\{ A \in \operatorname{Mat}(3,\mathbb{R}): A^tGA=G \}$ where ...
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48 views

How many normal planes?

Consider a surface $S$, a point $p$ on the surface, and the unit normal vector $\vec{N}$ passing through $p$ on the surface. There are infinitely normal planes passing $p$ and its unit normal vector ...
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92 views

Asymptotic invariants of infinite groups

I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept : $$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$ That is for ...
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40 views

Formalizing a proof using Germs to define a linear and injective map of the Algebraic Tangent of a manifold

I am trying to show that for X being an n-dimensional manifold and Y a k-dimensional manifold, U an open set and both $Y,U \subset X, p\in U$ there is a natural and well defined and injective map ...
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45 views

Exactness of $\Gamma^\infty$ Functor

Does anybody know a reference for fact that the Functor $\Gamma^\infty$, assigning to every smooth vector bundle $\mathcal{E}\to M$ the $C^\infty(M)$-module $\Gamma^\infty(\mathcal{E})$ of smooth ...
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97 views

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. ...
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69 views

Radius of geodesic disc

I'm trying to find the radius of a geodesic disc with center (0,0,1) on the paraboloid $$ r(u,v) = [\sqrt{1-u} \cos(v), \sqrt{1-u} \sin(v),u] $$ expressed by $u.$ That is, the length of the curve ...
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63 views

Operations on vector spaces

Is there a symmetrical analogue of the grassmann tensor products? Is it the polyadic tensor product? Are such symmetrical products used anywhere in differential geometry?
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69 views

Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
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80 views

(Basic) question regarding Einstein-Hilbert-functional / total scalar curvature

I got a question regarding the total scalar curvature / Einstein-Hilbert-functional. I found it in Peter Topping's book "Lectures on the Ricci flow" (you can download it for free here: ...
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77 views

Proof of Lie theorem on solvable Lie algebra

I am reading a book of Helgason. As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that $v$ is an eigenvector of any element of $g$. I can follow the proof in ...
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25 views

A better way to see this relation concerning Ricci tensor components

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
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78 views

What is the needed background to study tensors?

What is the needed background to study tensors? I do not plan to study it but, I want to know the background! It's a matter of curiosity . Thanks.
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240 views

Relation between triangle and its circumscribed circle, on the surface of a sphere. Generalizations to higher dimensions.

Consider the unit sphere in $R^3$ and an equilateral triangle of side length 1, with all vertices on the surface of the sphere. Now project (from the center of the sphere outward) the triangle and its ...
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59 views

How can I align the angle between points with the magnetic heading as the points move?

I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point. If I find the angles in the ...
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118 views

Deformation retract

How to prove that $r_t$ is a deformation retract $M^a=\lbrace q\in M ; f(q)\leq a\rbrace$ We have the definition : $r_t$ is a difformation retract if: $r_t$ is a continius ,onto application ...
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33 views

Equivalence class involving Lie Brackets..

Can anyone help me with a proof, I spent hours on it and nothing: I want to show $$\overline{[fX, Y]}=f\overline{[X, Y]},$$ where $X$ and $Y$ are smooth vector fields on a smooth manifold $M$, $f\in ...
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67 views

Cotangent bundle of a complex projective space

How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?
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54 views

coordinate transformation of the local pull back of the Maurer Cartan form

This questions asks how to express the pull back of the Mauere Cartan form on a Lie group to a smooth manifold. The Lie group,$G$, is the structure group of a smooth vector bundle over the ...
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42 views

Do sections defined in different patches give the same element in an associated bundle?

We can read here in p. 10 (penultimate equation) that the sections on associated bundles (not necessarly vector bundle) are defined by functions $f:P\longrightarrow F$ that must satisfy the ...
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48 views

Distances of points on geodesics

The setting: -Let $(M,g)$ be a complete Riemannian manifold and let $\pi:E \rightarrow M$ be its universal covering with the pullback metric. -Let $\alpha,\beta:[0,1] \rightarrow E $ be two minimal ...
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133 views

Energy functional

During my study on Ricci Flow I faced some functional known as enery functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...
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53 views

Cohomologies of $\mathbb R^n$ with rational differential forms

We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
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91 views

Geodesic on a Hilbert manifold

Given a Hilbert manifold $\mathcal H$ (always using the natural Hilbert inner product) and a geodesic $\Gamma(t)$ in this manifold, can one show that the projection of this geodesic onto a submanifold ...
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214 views

Riemannian curvature and its application on covariant derivative of tensors

This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
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49 views

Problem solution hint about boundary of boundary of chains from Arnold' book mathematical method

On his book Mathematical Methods of Classical Mechanics, (Chapter 7, Section 35, Problem 10), Arnold asks to show that the boundary of boundary of any chain is zero. He gives hint saying: by the ...
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111 views

Gelfand-Leray integral for forms with noncompact support

Let $\omega$ be a smooth $n$-form with compact support on domain $\Omega \subseteq{\mathbb{R}^n}$ and let $f \colon \Omega \to \mathbb{R}$ be a smooth function with nonvanishing differential. Then for ...
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68 views

Connection forms on manifolds in Euclidean space

This question comes from trying to generalize something that it easy to see for surfaces. Start with an oriented surface smoothly embedded in Euclidean space. The embedding determines two mappings of ...
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119 views

Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
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81 views

Torsion-free $G$-Structures

I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of ...
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97 views

Vector fields as section of tangent bundle

We can define vector fields on manifolds in two ways. The way I first saw was that a vector field was a linear map $C^\infty(M) \to C^\infty(M)$ satisfying the Leibniz rule (aka product rule). We can ...
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51 views

A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which ...
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60 views

Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov

Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
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81 views

Smoothing corners of a handle attachment

Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
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259 views

Difference between parallel transport and derivative of the exponential map

Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then we have to ways to map $T_pM$ to $T_{c(t_0)} ...
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120 views

Differential geometry textbook or lecture notes on the riccati equation and riccati inequality

I took a course on differential geometry and didn't get one specific topic well, so I am searching on some additional metrial to understand it in a better way. This wasn't a course about classical ...
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91 views

How to construct an orthogonal coordinate system from a smooth planar curve?

Given a planar curve $$\gamma:\mathbb R\to\mathbb R^2, t\mapsto \gamma(t) \text{, normalized to } |\gamma'(t)|\equiv1,$$ the tangential vector $$T(t) = \gamma'(t)$$ and the normal vector $$N(t) = ...
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86 views

Why does the map $x^2$ have constant rank?

I'm just trying to wrap my head around the rank of a map via some examples. Now, if I have the smooth map of manifolds $F:\mathbb{R} \to \mathbb{R}, F(p) = p^2$, then the differential is given by ...
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169 views

Difference between distance between two points and metric

if i have a line element given e.g. $ds^2=\frac{dx^2+dy^2}{2y} $ is it then always possible to derive a distance between two points in this metric? and how would one determine the length of a curve if ...
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34 views

How is integration of differential form defined as, and how to calculate it

How is integration of differential form defined as? And how does one calculate the value of integration?
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59 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
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84 views

Volume form on complex space, notation

If $\Omega = dz^{1}\wedge..\wedge dz^{n}$, $z^{1}$,..,$z^{n}$ are complex numbers. What does the notation $\Omega (x^{1}\wedge.. \wedge x^{n})$ means? $x^{1},..,x^{n}$ are real parts of ...
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107 views

Immersed Surfaces in Hyperbolic Space with Positive Intrinsic Curvature

Does anyone know of an example of a noncompact, immersed surface in hyperbolic 3-space with positive intrinsic curvature that is NOT embedded?
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76 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
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36 views

Nearly Kähler and special Kähler manifolds

We know that the most important example of a nearly Kähler manifold is the sphere $S^{6}$ and that $(\nabla_{X}J)Y=-(\nabla_{Y}J)X$ is valid in this case (J - an almost complex structure). Similar ...
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359 views

How to derive covariant derivative and Lie derivative of tensors

1) As title says, how does one derive the following equation for covariant derivate of tensor: $A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$ where ...
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35 views

Dual connections, examples

If $\overline\nabla^{*} $ is a dual connection of connection $\overline\nabla $, and we have the Gauss equations: \begin{align} \notag \overline\nabla^{*}_{X}Y=\nabla^{*}_{ X}Y + h^{*}(X,Y) ...
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37 views

Boundaries- regularity and local parametrization

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...