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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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 ...
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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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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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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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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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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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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
66 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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51 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?
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1answer
25 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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52 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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68 views

Generalization of the Frenet-Serret frame.

Consider $\alpha : I \rightarrow \Bbb R^n$ a differentiable curve, $\mathscr{C}^\infty$ and parametrized by arc-length, to make our lives easier. If $n = 2$, we have only one signed invariant, the ...
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27 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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40 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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26 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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29 views

Functional of Einstein tensor

What does this equal to, and how do I actually calculate this correctly? $$ \frac{\delta G_{ab}}{\delta g_{cd}}=? $$
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3answers
43 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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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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45 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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44 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 ...
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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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Derivative of the inclusion of a submanifold

I know there are other questions similar to this one, but I just want you to tell me if what I'm doing is rigth and how to improve it. The problem is the following: (I'm using the definitions by ...
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1answer
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$[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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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 ...
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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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33 views

Shape operator and orthogonality of eigenvectors

When studying differential geometry (at a hobby level) I always run into problems when it comes to the varying notations and statements about the shape operator ...
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1answer
21 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 ...
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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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35 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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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 ...
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252 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!
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2answers
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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
52 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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49 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 ...
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1answer
44 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 ...
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60 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 $G$ ...
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1answer
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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 ...
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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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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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2answers
278 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
55 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.) $\mathfrak{X}(M)$ ...
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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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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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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 ...
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Divergence Theorem To Calculate Surface Integral

$M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}\leq z$} We are asked to find the surface area of this surface. This is my way: $\partial M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}= z$} so the ...
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1answer
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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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1answer
33 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 ...
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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 ...
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1answer
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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
67 views

motivation for spaces with functions

Let $k$ be a field. A space with functions over $k$ is topological space X together with a family $O_X$ of k-subalgebras $O_X(U)\subseteq Map(U,k)$ for every open set $U$ that satisfy a) If ...