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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Is exponential map an immersion?

Let $M$ be a connected Riemannian manifold. For $p\in M$, the injectivity radius at $p$ is the sup of the $\epsilon >0$ such that the Riemannian-distance ball $B_\epsilon (p)$ is a geodesic ball, ...
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750 views

Basic understanding of a metric.

What is a metric ? Do a metric depend on the system of coordinates I use ? Does it depend on surfaces (or higher dimensional manifolds. Correct me if I'm wrong using the word) the coordinate frames ...
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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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39 views

Relation between differential geometry and differential geodesy

I am not exactly clear on what are the differences between differential geometry and differential geodesy. Are principles in differential geometry used in differential geodesy ? It appears that ...
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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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1answer
18 views

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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2answers
43 views

Examples of surfaces

I have to find an example of a surface of revolution excluding a sphere and a cone. Is $\sigma(x,y)=(\cos x, 5, x^2+y^2)$ such an example? $$$$ I also have to find an example of a surface the ...
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1answer
66 views

Lie groups where $x \mapsto x^2$ is a diffeomorphism?

In every Lie group $G$ the function $x \mapsto x^2$ is a local diffeomorphism in a neighbourhood of the identiy. (This is because its differential is: $v \mapsto 2v$ when considered as a map from ...
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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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1answer
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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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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1answer
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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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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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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1answer
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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1answer
62 views

Injective immersion that is not trajectory of any flow

Let $M$ be a compact manifold of dimension $m \geq 2$. Show that there exists an injective immersion of $\mathbb{R}$ in $M$, whose image is not the trajectory of any flow. I know how to do it for ...
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2answers
46 views

Tangent line of Lissajous curve?

I'm trying to find at how many points the tangent line of $(\cos(3t),\sin(2t))$ goes through the point $(3,0)$. My attempt: This is the same thing as saying for how many values of $t$ do we have ...
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1answer
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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1answer
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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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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1answer
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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1answer
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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0answers
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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2answers
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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1answer
29 views

Differential Geometry Proof Regarding Arclength, Tangents, Curvature, and Parameters

Consider a regular curve q(t) with arclength parameter s. Show that if $T(t_{n}) \neq T(t_{0})$ and $t_{n} \rightarrow t_{0}$, then $$1 = lim_{t_{n} \rightarrow t_{0}} \frac{|\theta(t_{n}) - ...
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1answer
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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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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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 ...
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0answers
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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1answer
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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1answer
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 ...
1
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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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0answers
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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0answers
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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votes
1answer
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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1answer
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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0answers
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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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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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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3answers
181 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
2
votes
1answer
35 views

Polyakov action in complex coordinates

Let $\Sigma$ be a compact $2$-manifold with riemannian metric $g$ and $f:\Sigma \to \mathbf{R}^n$ given locally by $f_1(x_1,x_2),\dots,f_n(x_1,x_2)$. Define $$ S(f,g) = ...
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1answer
67 views

Is this a misprint in Do Carmo's 'Curves and Surfaces'?

I'm reading the following section from the book 'Curves and Surfaces' by Do Carmo, but I'm stuck and after having gone over this like 10 times I'm starting to think it must be a misprint. The problem ...
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2answers
65 views

Which is the intersection?

I am looking at the last question of the following exercise: $$$$ Which exactly is the intersection of any surface from one family of the triply orthogonal system with any surface from another ...
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1answer
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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0answers
34 views

Is this subset of $\mathbb{R}^{3}$ a topological manifold?

Consider the set $\mathcal{M}_{1} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ y = -1 \ \}$, this is a plane. Also consider the set $\mathcal{M}_{2} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ x=y=0 \ \}$, which ...
2
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1answer
26 views

$X_z=\frac{d}{dt}_{|t=0} \Phi_t(z)$ has flow $\Phi_t$

Let $M$ be a manifold, $\Phi_t, t\in \mathbb R$ a one parameter group of diffeomorphisms and $X$ a vector field on $M$ definied by $$X_z:=\frac{d}{dt}_{|t=0} \Phi_t(z).$$ Show that $\Phi_t$ is the ...
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votes
1answer
116 views

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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2answers
79 views

$S^n$ admitting nowhere zero vector field implies identity map of $S^n$ is homotopic to antipodal map? [closed]

If $S^n$ admits a vector field which is nowhere zero, does it follow that the identity map of $S^n$ is homotopic to the antipodal map?
0
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1answer
26 views

Integral of a portion of a curve

I'm struggling with this question: It says: let $C$ be the portion of the curve $y=2 \sqrt{x} $ between $(1,2)$ and $ (9,6).$ Find $ \int_C3y \, ds$ Any clue that would assist me?