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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the analog to geodesic problem but with the area ?? for a variational problem

so we have that if we minimize the functional $$ S= \int_{a}^{b}\sqrt{g_{a,b}\dot x_{a}\dot x_{b}}$$ then the Euler Lagrange equations are $$ \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu ...
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25 views

Exterior derivative of a coordinate function

I'm starting to learn about differential forms. From what I understand the coordinate differential forms $dx^1, \dots, dx^n$ are actually the exterior derivatives of the coordinate functions $x^1, ...
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15 views

Uniqueness of integral curve

Given a vector field $X$ on a smooth manifold $M$ and a point $p \in M$, we know that there exist an open neighborhood $U$ of p, an $\epsilon >0$ and a unique local flow $F : U \times (-\epsilon, ...
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45 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
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44 views

Finding the equation of and drawing a cardiod

I have attempted drawing this many times and cannot come up with a cardioid. I believe the AM1 and AM2 are what is confusing me. I honestly have no idea where to start in order to find the equation. ...
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37 views

An analogue to the Koszul formula in the “wrong” degrees

Let $M$ be a smooth (closed, connected) manifold, $b\in\Omega^k(M)$, $P\in\Gamma(\Lambda^pTM)$ and $Q\in\Gamma(\Lambda^qTM)$ such that $p+q=k-1$. We denote by $[,]$ the Schouten-Nijenhuis bracket ...
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45 views

Differential of Sum of Two Functions is Sum of Differentials

Let $M$ be a smooth $n$-manifold and $f, g:M\to \mathbf R^n$ be smooth functions on $M$. Let $p$ be a point on $M$. I want to show that $d(f+g)_p=df_p+dg_p$ without passing to a chart about $p$. ...
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39 views

Straight line segment

I want to show that the straight line segment joining two points $p_1$ and $p_2$ in a plane is the shortest path between $p_1$ and $p_2$. I have tried the following: The straight line segment ...
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51 views

Relation between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?

Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the usual Lie algebra of vector fields on $M$; that is ...
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23 views

1-dim Vector Bundle sufficient condition to be trivial

I'm a physics student studying differential geometry. I'm trying to understand how vector bundles work, I have the following exercise. Let be $ L $ a $1$-dim vector bundle on $M$. Prove that if ...
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46 views

The limit of a uniform convergent sequence of isometries is an isometry (problem 6-3 of Lee's “Riemannian manifolds”)

I'm trying to prove the following theorem: let $f_n : M \to N $ a sequence of isometries of Riemannian manifolds that converges uniformly to a function $f:M \to N$: prove that $f$ is an isometry too. ...
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79 views

What is the space curve with curvature and torsion obeying

$ \kappa = \cos s, \tau = \sin s $ and passing through (1,0,0), TNB triad identity matrix? previous link When numerically computed it looks like a catenoid surface of revolution for all ...
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65 views

What is the formal definition of tangent hyperplanes?

My questions arise from 11.21 DEFINITON and 11.22 THEOREM ,both of them presented on p341 An Introduction To Analysis W.R.Wade 3ed. Question 1. Considering 11.21 DEFINITON, Let ...
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122 views

Why do these two facts imply that $S^n$ is not parallelizable for $n$ even, $n \ge 2$?

Consider the following two facts. If $S^n$ admits a vector field which is nowhere zero, the identity map of $S^n$ is homotopic to the antipodal map. For $n$ even, the antipodal map of $S^n$ is ...
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36 views

What does it allow to see Differential Geometry from an abstract viewpoint?

I am currently learning Differential Geometry from the "categorical" point of view. We use sheaves, ringed spaces, group objects,... to define smooth manifolds, vector bundles,... My previous course ...
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543 views

Lie algebra of normal subgroup is an ideal

I want to prove that if $G$ is a connected Lie group, $H$ is a normal Lie subgroup of $G$, $\mathfrak{g}$ and $\mathfrak{h}$ their respective lie algebras, then $\mathfrak{h}$ is an ideal of ...
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33 views

Normal coordinates and the metric matrix

While trying to follow and check the proof of Theorem 1 in this work on manifold averaging I reached the notion of normal coordinates. An important property is that the metric tensor at a point ...
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53 views

Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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32 views

Tensor Contraction Invariance

On page 86 of Bishop and Goldberg's Tensor Analysis on Manifolds you are asked to show that contractions are invariants. Rather than doing it awkwardly for an (r,s)-tensor (r is the contravariant and ...
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567 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
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41 views

Reference request: Proof that every product of vector space is isomorphic to the tangent bundle

On Wikipedia, it says On every tangent bundle $TM$, considered as a manifold itself, one can define a canonical vector field $V : TM → TTM$ as the diagonal map on the tangent space at each ...
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47 views

Möbius strip parameterization and charts

A parameterization of the möbius-strip is given by : $$\begin{align}M=\{ (x,y,z) \in \mathbb R^3: x &= \cos t(1+ s\cos(t/2)),\\ y &= \sin t(1+ s\cos(t/2)),\\ z &= s\sin(t/2), \\ t ...
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44 views

How to express the property “spiraling around” in differential geometry

I am starting to learn differential geometry, and reading the book "Differential Geometry of Curves and Surfaces" of Manfredo. The I got stuck on this problem: Let $\alpha(t) = ...
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46 views

Equalities and inequalities for quadrilaterals in hyperbolic space

In euclidean space any quadrilateral satisfies equalities and inequalities $$a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2$$ $$a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2$$ where $a,b,c,d$ are the side lenghts, ...
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42 views

Laplace-Beltrami of the Gauss map

I'm looking for the proof of very nice identity about the Laplace-Beltrami operator of the Gauss map $N$ of a regular surface in $\mathbb{R}^3$ given by a patch $X$. I want to show that $$\Delta N = ...
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1answer
44 views

Riemaniann metric is an element of?

We are given the definition: A riemaniann metric $g$, is a map: $g:p\rightarrow<.,.>|_{T_pM}$ where $<.,.>|_{T_pM}$ is the usual bilinear symmetric etc.. It also says that the metric ...
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49 views

Question about creating a volume form for $SL(2,\mathbb{R})$

This problem comes out of R.W.R. Darling (Differential Forms and Connections) ch.8. In the chapter he shows that if $M$ is an $n$-dimensional differential manifold immersed in $\mathbb{R}^{n+k}$, and ...
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69 views

Is the pairing induced by the wedge product and integration nondegenerate on de Rham forms?

Let $M$ be a compact, oriented, smooth $n$-manifold and let $\Omega^*_{\mathrm{dR}}(M)$ be the commutative differential graded algebra of de Rham forms on $M$. We can define a pairing: \begin{align} ...
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103 views

Taylor expansion - How can we deduce that?

We have that $$\tilde{E}=\frac{u^2}{r^2}+\frac{Gv^2}{r^4}, \ \ \tilde{F}=\left (1-\frac{G}{r^2}\right )\frac{uv}{r^2}, \ \ \tilde{G}=\frac{v^2}{r^2}+\frac{Gu^2}{r^4}$$ and that ...
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25 views

Intersection of a surface with a plane

I am looking at the following exercise: The line of striction is $\Gamma=\gamma-\frac{\dot\delta\cdot\dot\gamma}{\|\dot\delta\|^2}\delta$. For the first part I have done the following: We have ...
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82 views

Covariant derivative ambiguity

I'm studying general relativity and am running into an ambiguity with the covariant derivative. The covariant derivative acting on a scalar is, in a co-ordinate basis, simply $$\nabla_X f = X^a ...
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17 views

Curves with distance between them growing locally as $o(d^k)$

Context: I'm searching for some standard definitions related to order of contact between curves (and smooth manifolds in general). My research has taken me to the concept of jets. Simply speaking, a ...
65
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1answer
3k views

What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. [You might imagine ...
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1answer
196 views

Laplace operator in spherical coordinates, abstract approach

I'd like to show the well-known formula of the Laplacian operator for euclidean $\mathbb{R}^3$ in spherical coordinates: $$ \Delta U = \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial ...
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30 views

Tensors in a vector bundle with infinite-dimensional fibers

If $\xi = (\pi,E,M)$ is a vector bundle such that each fiber has finite-dimension, and $V=\Gamma(M,E)$ is the space of smooth sections, then there is an isomorphism between $(\otimes^r ...
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28 views

Quotient of a manifold

Suppose we have a manifold $M$, and a connected submanifold $N$. We can make the quotient $\frac{M}{N}$, which send $N$ to a single point. Now, there are known restrictions on $N$ such that ...
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62 views

How do we get the terms $E, \ F, \ G, \ L, \ M, \ N$?

I am looking at the following exercise: Show that a curve $\gamma (t) = \sigma (u(t), v(t))$ on a surface patch $\sigma$ is a line of curvature if and only if $$(EM − FL) \dot u^2 + (EN − GL) \dot u ...
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86 views

Tensor product for vector bundles is commutative, associative, and has an identity element?

How do I see that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element?
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1answer
41 views

The parameter curves are asymptotic curves

I am looking at the following exercise: Let $p$ be a hyperbolic point of a surface $S$. Show that there is a patch of $S$ containing $p$ whose parameter curves are asymptotic curves. Show that ...
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1answer
44 views

Condition to be conformal

I am looking at the following exercise: Let $\Phi : U \rightarrow V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$. Write $$\Phi (u, v)=(f(u, v), g(u, v))$$ where $f$ and $g$ are ...
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39 views

Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
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2answers
55 views

Repeated application of the gradient on a Riemannian manifold

While reading about Sobolev spaces on manifolds, I encountered the following notation regarding the norm in $H^k$: $\| \nabla ^k f \|_{L^2}$. There are two questions here: 1) What kind of object id ...
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29 views

Is the curl of a vectorfield expressable in terms of a Lie derivative?

I am loving differential forms and the general version of Stokes theorem. However, I am also fond of the intuitive pictures of my older physics days of $grad$, $div$ and $curl$. I recently stumbled ...
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Is a general smooth rescaling of a complete vector field itself complete?

$\newcommand{\Ga}{\Gamma}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\til}{\tilde}$ $\newcommand{\M}{M}$ $\newcommand{\ep}{\epsilon}$ $\newcommand{\brk}[1]{\left(#1\right)} $ ...
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29 views

Approximation of piecewise smooth curves with same-lenght smooth curves in Riemannian manifolds

Let $M$ be a Riemannian manifold, and let $\gamma : [a,b]\to M $ be a piecewise smooth curve. Then, using Whitney's theorems, it can be proved that $\gamma$ is homotopic (by a homotopy relative to $a$ ...
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91 views

How do we get the property (iii)?

I am looking at the following exercise: $$$$ I have shown that (ii) and (iii) togenther imply (i). Now I want to show the second part. We assume that (i) holds that that $\gamma$ is not a ...
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294 views

How can we find geodesics on a one sheet hyperboloid?

I am looking at the following exercise: Describe four different geodesics on the hyperboloid of one sheet $$x^2+y^2-z^2=1$$ passing through the point $(1, 0, 0)$. $$$$ We have that a curve ...
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1answer
29 views

How to understand it will sweep out a 2-dim manifold?

As red line in picture below, I really can't imagine why it will sweep out a 2-dim submanifold. Below picture is from the 9th page of John M. Lee's 'Riemannian Manifolds: An Introduction to ...
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31 views

Diffusion on a Boundaryless Manifold and Tesselation

Suppose we are dealing with diffusion on a boundaryless manifold $M$. When people use the finite-element method to find an approximate solution, I always see them use \begin{align} \int_{M^*} W \Delta ...
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66 views

Umbilics on the ellipsoid

Show that, if p, q and r are distinct positive numbers, there are exactly four umbilics on the ellipsoid $$\frac{x^2}{p^2}+\frac{y^2}{q^2}+\frac{z^2}{r^2}=1$$ What happens if $p$, $q$ and $r$ are not ...