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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86 views

How to define Surface Laplacian on the sphere with radius 1

The simbol $\nabla_s f$ appears in a problem of my homework, and my professor thinks it means $$\nabla_s f:= \nabla f - \hat{n}(\hat{n} \cdot \nabla f )$$ or $$ \nabla_s := (I - \hat{n}\hat{n}^T ...
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0answers
56 views

Preimages of smooth maps between manifolds

Suppose $M$ and $N$ are both compact, connected, oriented $m$-manifolds without boundary and $f: M \to N$ is smooth. What additional condition(s) must $f$ satisfy so that there exists at least one ...
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0answers
31 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
2
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1answer
26 views

two problems to finding the regular value of matrix group

$1$. Let $F:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ given by $F(X)=X^TX$. $2$.$F:M_2(\mathbb{R}) \to S_2(\mathbb{R})$ given by $F(X)=X^TX$ where $S_2(\mathbb{R})=$ {$X \in M_2(\mathbb{R}): X^T=X$}. ...
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2answers
45 views

A question on distributions

Let $\Delta$ be a smooth distribution on a smooth manifold $M$ and let $X,Y$ be 2 vector fields on $M$ which are tangent to $\Delta$ (namely $X(q),Y(q)\in \Delta_q\leq T_qM$ for every $q\in M$. I ...
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1answer
50 views

Definition of “Representing” a Handlebody (Lefschetz Fibration)?

Sorry, I could not find a clear explanation of the meaning of the word represented in the following:"any 4-dimensional 2-handlebody W can be represented by a topological (achiral) Lefschetz fibration ...
2
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1answer
31 views

A diffeomorphism between manifolds (or surfaces) that preserves the mean value of functions

Let $M$ and $N$ be two Riemannian manifolds with $f:N \to M$ a diffeomorphism with the following properties: for all $u \in H^1(M)$, $\hat u := u\circ f$ satisfies $\hat u\in H^1(N)$ and furthermore ...
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0answers
247 views

A particular method of pulling back a metric on a submanifold

Let $S$ be a $(n-1)$-submanifold of a $n$-manifold $M$ and that be a submanifold of $(n+1)$-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using ...
4
votes
1answer
58 views

Perelman's F-functional and its analysis

While going through the Kleiner and Lott notes "Notes on Perelman's papers", I encountered an argument that seems wrong to me, or (more likely) I do not understand something. It is about the ...
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0answers
22 views

Why does the exceptional Lie group $G_2$ have dimension 14?

In ''Compact manifolds with special holonomy" by D. Joyce, on p. 242, the group $G_2$ is defined to be the subgroup of $GL(7,R)$ preserving the $3$-form: $$ \varphi_0 := dx_{123} + dx_{145} + ...
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2answers
54 views

is knot type invariant under diffeomorphism?

Is it possible to have a diffeomorphism of $R^3$ which changes the knot type, for instance the image of a trivial knot is a trefoil knot?
1
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1answer
72 views

What does a derivative with respect to metric mean?

What's the difference between the derivative with respect to metric and the derivative with respect to one of the coordinates? $$\frac{\partial }{\partial g_{ab}} or \frac{\partial }{\partial g^{ab}} ...
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votes
3answers
3k views

Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
1
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1answer
32 views

Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$?

Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$ ...
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0answers
46 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
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0answers
34 views

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. ...
3
votes
1answer
57 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 ...
13
votes
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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0answers
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
558 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 ...
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0answers
59 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
47 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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0answers
616 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 ...
0
votes
0answers
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 ...
1
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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 ...
2
votes
0answers
26 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
269 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 ...
1
vote
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
votes
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 ...
1
vote
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
30 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
votes
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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0answers
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 ...
1
vote
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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votes
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
67 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
498 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 ...
0
votes
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
votes
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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votes
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 ...