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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Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
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1answer
54 views

Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
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1answer
258 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
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44 views

Reference for construction on Riemannian Manifolds

I would like to know a book or an article where the connected sum of Riemannian manifolds is explained with some detail. I roughly know how to do the construction but I want to be able to check a more ...
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0answers
28 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
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0answers
53 views

Riemannian Submerssion

I am reading John Lee's Riemannian Geometry Chapter 3, and I want to do some exercises. I think that I need some hints to solve the following: (Problem 3-8 of that book) Suppose $M$ and $N$ are ...
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0answers
30 views

Functional of Einstein tensor

What does this equal to, and how do I actually calculate this correctly? $$ \frac{\delta G_{ab}}{\delta g_{cd}}=? $$
9
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1answer
557 views

Geometric intuition for the Weingarten map

Parameterize a hypersurface $M$ by $r: \Omega \rightarrow \mathbb{R}^n$, and let $T_p M$ denote the tangent space at $p = r(u)$. We define the Weingarten map to be the linear map $L_p : T_p M ...
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3answers
49 views

The difference between a fiber and a section of a vector bundle

If $ E_x := \pi^{-1}(x) $ is the fiber over $x$ where $(E,\pi,M)$ is the vector bundle. And the section is $s: M \to E $ with $\pi \circ s = id_M $. This implies that $\pi^{-1} = s $ on $M$. So then ...
2
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0answers
55 views

Symplectomorphism Preserves Cotangent fibrations

Let $M$ be a manifold with local coordinates $x^1,\dots,x^n$ and $T^\ast M$ the cotangent bundle with induced coordinates $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ . Let $\alpha$ be the tautological one form ...
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0answers
46 views

Banach Tarski paradox in Minkowski space

The Banach - Tarski paradox states: 'A three-dimensional Euclidean ball is equidecomposable with two copies of itself.' My simple question is: Does this paradox hold his validity also in the Minkowski ...
21
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610 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
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21 views

How does the reduction of the frame bundle affect the tangent bundle

Let $M$ be a differential manifold and $F(M)$ its frame bundle. Suppose there is a reduction of the structure group of $F(M)$ from $GL(m,\mathbb{R})$ to the Lie group $H$ and let $F_{r}(M)$ be the ...
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1answer
27 views

$[V,fW] = f [V,W] + V(f) W $ Lie product

Some notation: Let $M$ be a smooth manifold and denote derivatives by $d$. For a vectorfield $V$ and $f \in C^\infty(M)$ we write $V(f)(p) = d_pf(v_p)$ where $v_p = V(p)$. Further $[V,W]$ is a ...
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0answers
25 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
7
votes
3answers
263 views

Interior product of differential forms

The interior product of a 2-form $\beta$ and vector field $X$ is defined by $(i_x\beta)(Y)=\beta(X,Y)$ where $Y$ is a vector field. This is the definition of a 2-form (and it's similar for a ...
2
votes
0answers
23 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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1answer
22 views

Being smooth homotopic relation: proof

Suppose we have an open set $U$ in the plane and two $\cal C^\infty$ paths $\gamma,\eta:[a,b]\to U$ with the same endpoints (i.e., $P:=\gamma(a)=\eta(a)$ and $Q:=\gamma(b)=\eta(b)$). We say that ...
4
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3answers
123 views

Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
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1answer
36 views

formula of square of the covariant derivative

I am stuck with the calculation of $(\nabla ^2 \beta)(X,Y,Z_1,\dots,Z_r)$. In the following, capital letters are arbitrary vector fields. Suppose $\beta$ is an $(r,0)$ tensor. Denote $(\nabla ...
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0answers
29 views

Sectional curvature of orbits generated by an isometric action

Let $G$ be a connected Lie group acting isometrically on a pseudo-riemannian manifold, $M$. I need a practical way to calculate the sectional curvature of the orbits at any point. Actually, I have ...
3
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2answers
255 views

Motivating differential geometry to high school students

What is the best way to motivate and explain what differential geometry to an audience of high school students? Any tips and suggestions are welcomed!
2
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2answers
25 views

Basic geometry proof about tetrahedron

Suppose tetrahedron ABCD such that AB is orthogonal to CD and AC is orgthongal to BD. Show that AD is orthogonal to BC. So i made a picture of a tetrahedron in 3 space and sort of look down at it ...
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1answer
54 views

Can every parameterised smooth curve be reparameterised by arc-length?

If someone can provide me a hint to a proof that would be awesome!
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130 views

Periodicity of Ideal Packings on Arbitrary Boundaries

This is a question that's been nagging me for some time that I find particularly interesting, but have no idea how to go about solving it. Consider some arbitrary solid $S$, such that $\partial S$ is ...
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2answers
284 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
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1answer
50 views

Background for 2 differential geometry questions

I encountered a couple of questions in a collection of differential geometry exams that I don't know how to approach. Of course I am NOT expecting a solution to these, but just a hint. If $S\subset ...
0
votes
1answer
45 views

Show that $T\mathbb S^1$ is diffeomorphic to $\mathbb S^1\times\mathbb R$

I just started learning Smooth Manifolds and got stuck on this question: Show that $T\mathbb S^1$ is diffeomorphic to $\mathbb S^1\times\mathbb R$ I can see that $T\mathbb S^1$ and $\mathbb ...
2
votes
1answer
56 views

Linear dual of vector fields

Suppose that $M$ is a smooth manifold and $\mathfrak{X}(M)$ is the set of smooth vector fields on $M$. There are basically two different linear structures on $\mathfrak{X}(M)$: 1.) $\mathfrak{X}(M)$ ...
3
votes
0answers
66 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
4
votes
1answer
497 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
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1answer
20 views

How do I analyze the partial derivative of the following summation?

I'm taking a course in Machine Learning where the Gradient Descent algorithm is being used for optimization. I'm in high school and I have a decent knowledge of both Differentiation/Partial ...
2
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0answers
20 views

Generalizations of the product neighborhood theorem

As far as I know, the Product Neighborhood Theorem for smooth manifolds states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization ...
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0answers
30 views

Finding curve that minimizes an integral due to constraints

In the euclidean plane I want a smooth curve $\gamma (t)$ which satisfy:$$\gamma (0)=(0,0)\quad\quad \gamma '(0)=(1,0)\quad\quad n\in \mathbb{N}\land n\neq 1\Rightarrow \gamma ^{(n)}(0)=(0,0)\\\gamma ...
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0answers
9 views

Holonomy group of codimension 1 foliation

This is the Ex2.29(2) in the book Introduction to Foliations and Lie Groupoids by : I. Moerdijk / J. Mrcun Let F be a foliation of codimension 1 with only compact leaves, then the holonomy ...
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votes
1answer
34 views

Tangent plane at a common point to two closed regular surfaces in ${\mathbb R}^3$.

I have two closed regular surfaces of class $C^2$ in ${\mathbb R}^3$, say $S_1$ and $S_2$, with $S_2$ enclosing $S_1$ and $S_1 \cap S_2\neq \emptyset$. I don't know how to justify the following: For ...
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0answers
23 views

normalized mean curvature flow with convex initial hypersurface has finite velocity

I can't understand the prove in [Xi-Ping Zhu] Lectures on mean curvature flows. The statement as follow. Lemma 3.5 (page 32) There exists a positive constant $C$ such that ...
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44 views

How to improve the isometric immersion of a $n$ dimension conformal metric of one variable conformal factor to be less than $2n-1$ dimensions?

Given a conformal metric $ds^2 = \omega(x_n)(dx_1^2+dx_2^2+\cdots + dx_{n-1}^2+ dx_n^2)$ in $\mathbb R^n$ with conformal factor of one variable $\omega(x_n)$, does there exist an isometric immersion ...
1
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1answer
45 views

Existence of a submanifold

Is it correct that if the bracket of two vectors $A_{1}$ and $A_{2}$ equals zero, then a submanifold tangent to $A_{1}$ and $A_{2}$ exists?
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votes
1answer
35 views

Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
2
votes
1answer
51 views

Is injection from manifold to tangent manifold well defined?

I would like to know if the injection map $i : M\to TM$, given by $i(x)\mapsto (x,0)$, is a well-defined and canonical application (not dependent on any particular coordinate chart).
2
votes
0answers
180 views

Taylor expansion of a vector field on manifold

In my work I have a need for some kind of analogue of Taylor expansion of a vector field on Riemannian manifold $\mathcal{M}$. I came to such an expression: $$ F(\operatorname{exp}_p(v)) = ...
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0answers
72 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
3
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1answer
183 views

Questions about flowing curves

A closed smooth non-self-intersecting curve in $\mathbb{R}^2$ having its curvature less than one is called a flowing curve. The three connected questions arise: How to prove that a disk of radius ...
2
votes
2answers
55 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
0
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1answer
28 views

curve of constant curvature on unit sphere is planar curve?

I've studied differential geometry and get this question. I'd like to verify following statement. curve of constant curvature on unit sphere is planar curve I've struggled with Frenet-Serret ...
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1answer
68 views

When is $x\mapsto |x|^{s-1}x$ a diffeomorphism?

Consider the function $f:B^n\rightarrow B^n$ from the disk to itself $$f(x)=\vert x\vert^{s-1}x$$ where $s>0$ and we are considering the euclidean norm (we define the function to be $0$ in the ...
7
votes
1answer
83 views

Ellipticity of Ricci tensor, does it depend on coordinates?

Well, I am afraid this is a silly question because I know the answer must be 'yes, it does'. But I don't see why. I put the problem in context. The ricci tensor can be regarded as a differential ...
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vote
2answers
84 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
2
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0answers
36 views

Compute the (surface) curvature

Consider the manifold $P:=S^2 \times \mathbb{R}^2$ equipped with the product metric $g((x,y),(x',y'))=g_{S^2}(x,x')+g_{\mathbb{R}^2} (y,y')$ where $S^2$ has a constant curvature that is 1. Let ...