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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A question about the boundary points of manifolds.

A topological $n$-manifold is a second countable Hausdorff space such that every point has a neighbourhood which is homeomorphic to an open ball centred at the origin in $\Bbb{R}^n$. The "centred at ...
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139 views

Manifold has uncountable many smooth stuctures if it has one

This is the Problem 1-6 of John Lee's Introduction to smooth manifold: Let $M$ be a nonempty topological manifold of dimension $n\geq1$. If $M$ has a smooth structure, show that it has uncountably ...
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82 views

Parallel Transport of Geodesic Velocity Vectors

Given a Riemannian manifold $M$ with Riemannian metric $g_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ and distance $d:M\times M\rightarrow\mathbb{R}$ determined by length of minimizing geodesics, ...
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34 views

Why are the charts in this set all $C^\infty$-related?

I'm reading the first volume of Spivak's differential geometry series, and am having a tough time convincing myself of something mentioned in the proof of Lemma 1, Chapter 2. Let $M$ be an ...
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61 views

An Atlas for $\mathbb{R}/{2\pi \mathbb{Z}} $

I've been having some difficulty finding an atlas for $\mathbb{R}/{2\pi \mathbb{Z}}$. The way I have been thinking of this so far is by using the standard projection map $\pi$ on open intervals of ...
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57 views

n-form density weight of the levi civita tensor

Why does the n form of levi civita $\epsilon:$=$w_{1}\wedge w_{2} ...\wedge w_{n}$ have a density of weight -1 where $w_{i}$'s are cobasis of some vector basis? I thought this geometrical quantity ...
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62 views

Tangential Vector

I have trouble to understand the following definition. If $M$ is a differentiate manifold then the tangent vector in $p\in M$ is a map $v:\mathcal{F}\rightarrow \mathbb{R}$ such that $v(f)=0$ if ...
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51 views

Law of Sines Non-Euclidean geometry

Is the following Law of Sines valid on all surfaces isometric to a sphere? $$\frac{\sin A }{ \sin a }= \cdots = \frac{ \sin C }{ \sin c } = E.$$ And similarly, Is the following Law of hyperbolic ...
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52 views

degree of determinant line bundle of pullback in terms of Chern class

I have a holomorphic map $f: C \to X$ where $\dim_\mathbb{C}C=1$ and $\dim_\mathbb{C}X=2$. Why is $\text{deg}\det f^*TX=-c_1(X)\cdot[C]$? I'm really curious to how that negative sign shows up.
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Is the natural map $\Omega^*(M)\otimes \Omega^*(F) \rightarrow \Omega^*(M\times F)$ injective?

As I self-study Bott and Tu 's Differential Forms in Algebraic Topology, I am stopped by some quite basic questions: 1) Let $M$ and $F$ are real manifolds, and let $\pi : M\times F \rightarrow M$ and ...
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89 views

Implicit Function Theorem (Two Variables)

While reading through a paper I came a cross a result (due to the Implicit Function Theorem) that I cannot derive. Let $u(x,y)$ be the solution of a PDE ($x$ and $y$ are independent variables). We ...
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123 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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52 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 ...
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30 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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74 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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61 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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85 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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34 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 ...
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41 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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59 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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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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34 views

Proving that a function is a $C^\infty$ submanifold in $\Bbb{R}^2$ of dimension 1

We need to prove that for all $c\in\Bbb{R}$ the set $\{x\in\Bbb{R}\,\colon\, g(x)=c\, \}$ is a $C^\infty$ submanifold ($g\,\colon\,\Bbb{R}^2\rightarrow \Bbb{R};(x_1,x_2)\mapsto x_1^3-x_2^3$) in ...
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52 views

Relation between principal bundle automorphisms and maps in $\Lambda^0(M,Ad(P))$

Let $ G \hookrightarrow P \xrightarrow{ \pi } M $ be a principal $ G $-bundle over $M$. Denote by $\mathrm{Ad} (P) = P \times _{ \mathrm{Ad} } G $ the non-linear adjoint bundle. It seems to me that ...
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29 views

How to create a simple closed curve homotopic to the trefoil knot?

How to create a simple closed curve that is homotopic to the trefoil knot $\overrightarrow{\alpha} (t)= \left ( \left (3+ \cos (3t) \right) \cos (2t),\left (3+ \cos (3t) \right) \sin (2t),\sin (3t) ...
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29 views

Any elementary derivation of the Pfaff integrability condition?

Suppose in $\mathbf{R}^N$ we have a one-form field, $ \theta = \sum_{i=1}^N \theta_i d x_i $. The Pfaff integrability condition is $d \theta \wedge \theta = 0$. Is it possible to give an ...
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42 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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48 views

Stoke's theorem explanation

Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one. I'll just state what I understand from it and I'd like someone ...
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25 views

Citing a result on obstruction to Lagrangian Embedding

Let $L$ be a closed orientable Lagrangian embedding in $\mathbb C^n$. Then $\chi(L) = 0$, where $\chi(L)$ denotes the Euler characteristic of $L$. This fact is more or less stated in section 3.2 of ...
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30 views

arbitrary reparametrization

Let $\alpha: (a,b)\rightarrow \mathbb{R}^n$ of class $C^{\infty}$ with $\Vert\alpha^{\prime}\Vert>0 $ then if $\{ k,m,n\} \subset \mathbb{R}_+$ there repametrizacion $\beta: (m,n)\rightarrow ...
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30 views

Equivalence between pullback connections of smoothly homotopic maps

Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we ...
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92 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
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63 views

All differentiable functions on $\mathbb{S}^n$

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
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34 views

prove that ''pullback'' maps forms to forms

Suppose we have $A: V_{1}\to V_{2}$ where $V_{1},V_{2}$ are real vector spaces. Then $A^{\star}:\mathcal{J}^{k}(V_{2})\to \mathcal{J}^{k}(V_{1})$ where $\mathcal{J}(V):=\{\text{space of all ...
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26 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
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95 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
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50 views

integral of a 2-form over an oriented manifold

An old exam question: Let $M$ be oriented submanifold of $\mathbb{R}^{n}$ of dimension $k$, let $\omega$ be a $k$-form on $M$. 1) Define $\displaystyle\int_{M}\omega$ 2) let ...
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30 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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39 views

Motivation for tensor density

Wiki has provided the basic definitions of the tensor density, but what I really want to know is the motivation and the advantage of this concept. Could anyone give me some ideas?
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56 views

Commutative differential graded algebra

The situation is the following: Let $V$ be a Lie algebra of finite dimension, say n, and let be the graded commutative algebra $(\bigwedge^{\bullet}V)^*:=(\bigoplus_{k=0}^n\bigwedge^kV)^*$ and define ...
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52 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
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48 views

Invariants of shape operators?

Let $S:V\longrightarrow V$ be a Linear Transformation, then the Characteristic polynomial of $S$ and therefore its Coefficient are invariant. Except the first and the last Coefficient that we know ...
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52 views

Differential Geometry and Origami

Would anyone know how to relate origami with differential geometry? I mean clearly you can see how geometry plays into it but how would you describe it in terms of differential
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32 views

Chain rule for tensor of family of tensor fields

Let $f_\tau$ be a $\mathbb R$-family (parameter $\tau$) of diffeomorphisms that map from $\mathbb R^4$ to $\mathbb R^4$. $f^*_\tau$ is the corresponding pullback (I think that is the correct term). ...
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65 views

curvature$=0$ implies straight line?

The fundamental theory of differential geometry states that: If there is a given curvature $\bar{\kappa}(s)>0$ and torsion $\bar{\tau}(s)$ which both of them are differentiable and continuous in ...
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82 views

principal axis of a volume from moments of inertia

I'm trying to calculate the expression to find the principal axis of a volume via its moments. In the 2D case I can formulate the problem by expressing the moments around arbitrary axes $x' = x \cos ...
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27 views

$[D,D']$ where $D$ is a derivation and $D'$ is skew

This is a proposition in 33 page of Foundation in Differential Geometry - KN I need some detail. Let $D^r(M)$ be a set of $r$-form. Then derivation (resp. skew-derivation) of degree $k$ is a ...
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36 views

Component of a pushover vector by one-parameter transformation

I am curious about a step on the proof that shows Lie derivative of a vector field is equivalent Lie bracket. Following comes from Nakahara. We define integral curves by vector field X and Y as ...
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38 views

Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
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197 views

Showing that two surfaces are not isometric/locally isometric

I am trying to solve an exercise which asks to show that two surfaces are not isometric and additionally that they are not locally isometric. The two surfaces presented are graphs. I know that if two ...
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37 views

Covariant Derivative to study Holonomy

Can someone provide an explicit definition to the $\mathbb{R}^{3}$-covariant derivative? For instance, I don't understand the following calculation. Let $$E_1 = (-\cos(u) \sin(v), - \sin(u) \sin(v), ...