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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57 views

$\omega$ is $1$-form on $S^1$.

Let $h: \mathbb{R} \to S^1$ be $h(t) = (\cos t, \sin t)$. How do I show that if $\omega$ is any $1$-form on $S^1$, then$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
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86 views

Proving Sard's theorem

Theorem: (Sard) Let $f:U\to \mathbb R^p$ be a smooth map with $U$ open in $\mathbb R^n$, and let $C$ be the set of all critical points of $f$. Then $m([f(C)]=0$ where $m$ is the Lebesgue measure of $\...
0
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1answer
57 views

Function never equal to 0

Let $p,q,x,y:(a,b)\to\mathbb{R}$ be $C^1 ((a,b))$ functions. Knowing that for $(u,v), (\tilde{u},\tilde{v})\in (a,b)\times (0,1)$ we have that: $$\begin{cases} p(u)+vx(u)=p(\tilde{u})+\tilde{v}x(\...
2
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1answer
43 views

Is level set near a maximum value a circle?

Let $f$ be a $C^2$ function defined on $[0,1] \times [0,1]$. Let $0 \leq f(x) < 1$ on $[0,1] \times [0,1] \setminus \left(\frac{1}{2},\frac{1}{2}\right)$ and $f(\frac{1}{2},\frac{1}{2})=1$. It has ...
2
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1answer
155 views

What does it mean to “calculate in local coordinates” on a manifold?

In differential geometry textbook one sometimes reads "calculating in local coordinates, we obtain..." What does this expression mean? Say, $M$ is a smooth manifold and $h$ is a function on $M$; what ...
2
votes
1answer
75 views

Holonomy reduction from constant spinors

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin, manifold, and let us denote by $S$ the corresponding spinor bundle. The Levi-Civita connection $\nabla$ on $(M,g)$ lifts to a unique spin ...
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174 views

Exercise 3.3 Riemannian Manifolds an Introduction to Curvature

STATEMENT: Let $\gamma(t)=(a(t),b(t)),t\in I$(an open interval), be a smooth injective curve in the $xz$-plane, and suppose $a(t)>0$ and $\dot{\gamma}(t)\neq 0$ for all $t\in I$. Let $M\subseteq \...
0
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1answer
47 views

Enneper surface is not injective

I'm having trouble proving the following statement: $x(u, v) = (u − u^ 3/ 3 + uv^2 , v − v^ 3/ 3 + u^ 2 v, u^2 − v^ 2 )$ is a minimal surface and x is not injective Proving that $x(u,v)$, ...
0
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52 views

Manifold arising from particular proof of Hairy Ball Theorem

Background, aka considerations to find my actual question In Geometry three, at the end of the last lesson, we sketched a proof of the famous Hairy Ball Theorem. The proof goes as follows. Lemma: ...
3
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1answer
80 views

Prove geodesics are straight lines if Riemann tensor is identically zero.

Suppose that $R^{a}_{bcd}\equiv 0$ in all of our manifold $M$ (in which we assume zero torsion). Prove that all geodesics are straight lines. I tried using Ricci's identity: $X^i_{;jk}-X^i_{;kj}=R^i_{...
1
vote
1answer
34 views

Obstruction to the existence of constant-rank sections of $T^*M\odot T^*M$

If $\alpha$ is a section of $T^*M\odot T^*M$, where $M$ is a smooth manifold, the rank of $\alpha$ at $m\in M$ is the codimension of the kernel of $\alpha_m$, i.e. the subspace of vectors $v_m\in T_mM$...
2
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1answer
64 views

If a composition of two maps is smooth, as well as one of the maps, then so is the other.

Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth. If any of $f$ and $g$ is smooth can we conclude that the ...
7
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1answer
148 views

Is $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ an exact form?

Let $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$. Then $$\int_{\mathbb{S}^2}\omega=\int_{\mathbb{S}^2}\sin\varphi\,\mathrm{d}\theta\,\...
2
votes
1answer
77 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
2
votes
1answer
187 views

Particular function in proof of flow box theorem

Flow Box Theorem If $M$ is a manifold of dimension $n$ and $X$ is a vector field on $M$ such that for a certain $p\in M$ $X(p)\neq0$, then there exists a chart $(U,\phi)$ on $M$ such that $p\in U$...
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0answers
49 views

Basic question: Riemannian Curvature is nondegenerate

$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$ is the curvature with respect to the Levi-Civita connection $\nabla$ of a metric $g$ on a manifold $M$. Define the Riemann curvature ...
9
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2answers
189 views

Good video lectures in Differential Geometry

I was not fortunate enough to learn Differential Geometry during my Masters. As now I am having my thesis in PDEs, and I miss a lot of mathematics from the people who do PDEs on Manifold setting. I ...
3
votes
1answer
247 views

Prove that the normal to a quadratic curve passes through a specific point

I've been asked to prove that the normal to the curve $y=2x^2 - 3x^{-1/2}$ at the point $(1,-1)$ passes through the point $(12,3)$. $\frac{dy}{dx} = 4x + \frac{3}{2}x^{-3/2}$ Hence, at the point $(1,...
2
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0answers
90 views

Flow of a left invariant vector field on a Lie group equipped with left-invariant metric and the group's geodesics

I think the answer to my question is known to many other people, but I'm still getting confused. Let $G$ be a (possibly infinite dimensional also) Lie group and $g$ be its Lie algebra. Consider the ...
5
votes
1answer
133 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
2
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2answers
119 views

The difference between the algebraic torus and the geometric torus

I know that the donut-shaped geometric object in $\mathbb{R}^3$ is homeomorphic to a square with identified opposite sites. However, while the latter has a clear symmetry between two dimensions, the ...
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1answer
116 views

Covariant derivative (or connection) of and along a curve

Let $c:[a,b] \rightarrow M$ be a curve parametrized by arc-length. Then $c': [a,b] \rightarrow TM$ such that $c'(t) \in T_{c(t)}(M).$ So essentially, I want to understand how $\nabla_{c'}c'$ is ...
3
votes
1answer
97 views

Exercise 2.3 Lee's Riemmanian Manifolds

Statement: Suppose $M\subseteq \tilde{M}$ is an embedded submanifold. a)If $f$ is any smooth function on $M$, show that $f$ can be extended to a smooth function on $\tilde{M}$ whose restriction to $M$...
2
votes
1answer
92 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 $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
votes
1answer
15 views

Express covariant transformation conveniently

Let $\omega = \sum_i \omega_i dx^i = \sum_{i} \nu_i dy^i$ be a 1-form in two different bases. Now, $(\omega_1,...,\omega_n)$ transform covariantly to $(\nu_1,...,\nu_n).$ My question is, can we ...
2
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1answer
101 views

Showing that an equation of a curve in the plane defines a surface in $R^3$.

A generalized cylinder is a ruled surface for which teh rulings are all Euclidean parallel. Thus there is always a parametrization of the form $$\mathbf{x}(u,v)=\beta (u)+v\mathbf{q} \; (\mathbf{q}\...
1
vote
1answer
59 views

Poincaré lemma and conservative vector fields

Let $U$ be some contractible neighbourhood of $0\in\mathbb{R}^n$ and let $X=\sum_{i=1}^nX_i\frac{\partial}{\partial x_i}$ be a (smooth) vector field on $U$. This vector field can be thought as a ...
1
vote
1answer
45 views

Poincare: Change of form to a primitive 1-form

Given a 2-form w on $R^3$, such that dw=0; how does one find all 1-forms k such that dk=w? Provided a homotopy H(t) one can pull back w and apply the Poincare operator on it. But what if one does not ...
0
votes
1answer
25 views

Deduce that there is a cup product that is well-defined

I have showed that if $\alpha$ and $\beta$ are closed forms on a smooth manifold $M$, then $\alpha \wedge \beta$ also be closed. Further, if one of $\alpha$ or $\beta$ is exact, than $\alpha \wedge \...
1
vote
1answer
33 views

Smooth function from function with singularity

Having an application $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$ in $C^{\infty}((t_0-\varepsilon, t_0+\varepsilon))$ with $f(t)=0\Leftrightarrow\ t=t_0$ and knowing that: $\exists\ \lim\...
1
vote
1answer
34 views

Globally defined exponential in a particular homogeneous space

I'm currently working in a particular conformal compactification/completion of the Minkowski space-time, but I'm stuck at showing that the exponential of every vector field in it is globally defined. ...
2
votes
1answer
51 views

Is that application smooth?

Having an application $f:(t_0-\varepsilon, t_0+\varepsilon)\backslash\{t_0\}\to\mathbb{C}$ in $C^{\infty}((t_0-\varepsilon, t_0+\varepsilon)\backslash\{t_0\})$ with $|f(t)|=1,\forall t\in (t_0-\...
1
vote
1answer
53 views

Degree of smooth maps are equal $\Rightarrow$ homotopic

It is an easily proven theorem that if $f,g:M\to N$ are smooth maps that are homotopic maps between compact, connected, oriented, smooth manifolds of dimension $n$, then $\deg f=\deg g$. I was ...
1
vote
1answer
27 views

Conditions on $a,b,c,d$ such that $\gamma (t)$ is regular for all $t$?

I solved an exercise in my book and I was wondering if someone could look at my answer and tell me if it is correct please? The exercise is this: Let $\gamma (t) = (a \cos t + b \sin t, c \cos t + ...
3
votes
0answers
47 views

Is this a normal form for $4$-forms on manifolds?

Starting from a $2$-form $\omega$ which is nondegenerate and closed on a $2n$-dimensional manifold, it is always possible to find local coordinates $x_1,y_1,\ldots,x_n,y_n$ so that the form $\omega$ ...
5
votes
0answers
199 views

Second Variation of Area Functional

This is a follow up question from this one. I have proved that given a parametrized surface ${\bf x}$, the mean curvature is zero if and only if it is a critical point of the area functional. Then ...
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vote
0answers
29 views

How can I get this new Gaussion curvature and mean curvature?

Let $M$ be a surface in $\mathbb{R}^3$ oriented by $\vec n$. For a fixed number $\varepsilon\neq0$, let $F:M\to\mathbb{R}^3$ be a mapping such that $F(p)=p+\varepsilon\vec n(p)$ and denote $\overline{...
1
vote
1answer
46 views

Diffeomorphism between vector bundles

I have some difficulty solving the following problem: Let $M$ be a diffentiable manifold of dimension $m$, which admits a global base of differentiable vector fields $\{X_1,\ldots,X_m\}$; this $\{X_1(...
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40 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E \...
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26 views

Prove that orthogonal n x n matrices form a $C^1$ surface of dimension n(n-1)/2 in $\mathbb{R^n}^2$ [duplicate]

Consider the function $F:Mat_n → Sym_n$ defined by the formula $F(A) = A∗A$. $Mat_n$ denotes the vector space of n × n matrices with real entries, while $Sym_n$ denotes the vector space of symmetric $...
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0answers
38 views

Imagening the Thurston geometries

I can (more of less) imagine how it would look if space was Euclidean, spherical of hyperbolic. But there are 8 Thurston geometries see https://en.wikipedia.org/wiki/Geometrization_conjecture how ...
7
votes
1answer
204 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why $\nu_{\...
2
votes
0answers
41 views

Wedge product of Lie algebra valued differential forms [duplicate]

Let $\mathfrak{g}$ be the Lie algebra of a matrix Lie group. Furthermore, let us consider the following $\mathfrak{g}$-valued $p$-form and $\mathfrak{g}$-valued $q$-form: \begin{equation} \begin{array}...
13
votes
2answers
369 views

I don't get the relationship between differentials, differential forms, and exterior derivatives.

I don't get the relationship between differentials, differential forms, and exterior derivatives. (Too many $d$'s getting me down!) Here are the relevant (partial) definitions from Wikipedia; ...
3
votes
2answers
96 views

Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
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1answer
64 views

Is my graph a tree?

Let M be a smooth connected manifold. G is a group act on M cocompactly and suppose there is a harmonic function $h$ on M with minimal energy.$h:\rightarrow [0,1]$ such that h is nonconstant and ...
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votes
1answer
48 views

How to get the result in this way?about determinant [closed]

In theory of surfaces using the fact that $\vec v \in T_p(M)$=$\alpha\vec x_u+\beta\vec x_v$ and $ S_p(\vec v)\times \vec v=\vec 0, $ deduce that a nonzero tangent vector can be a principal ...
2
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1answer
54 views

Explaining problem in Gadea's “Analysis and Algebra on Differentiable Manifolds”

I have a lot of trouble trying to explain to myself what the author did in problem 1.102 (the answer is in the link): Let $TM$ be the tangent bundle over a differentiable manifold $M$. Let $\...
3
votes
1answer
65 views

How to show the following vector bundles are equivalent?

Given a smooth sub-manifold $X$ of $\mathbb{R^n}$ and define the diagonal in $X \times X$ to be $$\triangle = \{(x,x) \mid x \in X \} \subset \mathbb{R^n}\times \mathbb{R^n}$$ and normal bundle to $\...
3
votes
1answer
45 views

Defining functions for connected sets

Let $\Omega \subset \mathbb{R}^n$ an open, bounded and connected set with a $C^2$ boundary and a function $\rho \in C^2(\mathbb{R}^n)$ such that $$ \Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 \},\...