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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47 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 ...
1
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1answer
31 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 ...
1
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1answer
24 views

Degree of map on $U(n)$ and roots in $U(n)$

Recently I went to a talk of A.Thom in which he sketched a proof of the fact that the groups U(n) satisfy the Kervaire-Laudenbach conjecture. At some point in the proof you have to argue that the map ...
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1answer
42 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 ...
0
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1answer
11 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 ...
8
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2answers
73 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 ...
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0answers
40 views

Is this differential 2-form closed

Consider a unit sphere $S^2 \subset R^3$ and a map $\omega_p : T_pS^2 \times T_pS^2 \to \mathbb{R}$ defined by $$\omega_p(u,v) = (u \times v) \cdot p$$ How do I know is this 2-form (on $S^2$) closed ...
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vote
1answer
20 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 ...
0
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1answer
27 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} \; ...
1
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1answer
35 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 ...
3
votes
1answer
52 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 ...
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1answer
20 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
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1answer
27 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\ ...
2
votes
1answer
43 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 ...
2
votes
1answer
69 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 ...
1
vote
0answers
18 views

Pipe-fitting conditions in 3D

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a pipe of diameter $D$ around it. Questions: What are the set of conditions ...
1
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1answer
26 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 ...
3
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0answers
37 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 $\omega$ ...
1
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0answers
28 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 ...
1
vote
1answer
35 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 ...
1
vote
1answer
29 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 ...
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0answers
24 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
30 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 ...
5
votes
1answer
45 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 ...
0
votes
0answers
19 views

Exponential map and convergence

Suppose that $M$ is smooth compact manifold and let $y \to x$. Let also $f \in C^{\infty}(M)$ be a smooth function. I consider the expression $\exp_y^{-1}(x)(f)$: then it follows that it converges to ...
3
votes
2answers
55 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 ...
10
votes
2answers
128 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
1answer
50 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 ...
2
votes
0answers
32 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 ...
2
votes
1answer
36 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 ...
7
votes
2answers
83 views

Top degree de Rham cohomology determines an orientation

Let $M^n$ be a smooth, compact, orientable, connected manifold. We know then that $H^n_{dR}(M^n)\simeq \mathbb{R}$ by the map $[\omega]\mapsto \int_{M^n} \omega$. I was wondering if, given an ...
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vote
1answer
26 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. ...
1
vote
1answer
30 views

Equation of an osculating plane to a curve (2 variables)

Maybe I am just reading this wrong, but here goes: Find the equation of the osculating plane to the curve \begin{cases} x = a\cos(\theta) \\ y = a\sin(\theta) \\ z = a(1+\sin(\theta)) ...
5
votes
2answers
55 views

Curvature tensors and bivectors

At the beginning of the paper "The curvature of 4-dimensional Einstein spaces", by Singer and Thorpe, the authors define the space $\mathcal{R}$ of curvature tensors of the vector space $V$ as the set ...
2
votes
1answer
32 views

Calculate the pushforward of smooth map between manifolds

Let $\Phi : GL(n)\rightarrow Sym(n)$ be defiened by $\Phi (A)=AA^T$. I can not see how to get the "right" pushforward, I.e I want help in understanding the pushforward $\Phi _*:M_I(n)\rightarrow ...
1
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1answer
30 views

Codifferential and corresponding homology theory

This is the kind of a natural question which can come to mind after completing the standard course in differential geometry and homology theory: lety us start with a smooth manifold $M$. One can ...
0
votes
1answer
42 views

How to get the result in this way?about determinant

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 ...
3
votes
1answer
27 views

The set of all critical points of a smooth map is closed

Let $f : \mathbb{R}^m \to \mathbb{R}^n$ be a smooth map. How do I show that the set of all critical points of $f$ is closed in $\mathbb{R}^m$? (Here, a critical point is a point $x \in \mathbb{R}^m$ ...
0
votes
1answer
52 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 ...
1
vote
0answers
50 views

How to use chain rule on it?

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 ...
2
votes
3answers
75 views

What does it mean that we can diagonalize the metric tensor

On a Riemannian manifold $M$, the matrix representation is diagonalisable, cause the tensor is symmetric. What is the physical meaning behind this? I mean, in Riemannian geometry, we always get a ...
6
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0answers
85 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 ...
2
votes
0answers
19 views

Orientation at the boundary for manifold with corners: the simplex

Consider the $n$-simplex $$\Delta[n]:=\{(t_{1},\dots,t_{n})\in \mathbb{R}^{n}\: : \: 0\leq t_{1}\leq t_{2}\leq \dots \leq t_{n}\leq 1\}.$$ This is a manifold with corners. The cofaces map $d^{i}\: : ...
2
votes
1answer
77 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
6
votes
1answer
66 views

When vectors act on scalars.

Background. I've been struggling through an introduction to differential geometry this semester. Recently, a tiny part of what we've been learning "clicked" for me, and to solidify this, I'd like ...
0
votes
0answers
36 views

How to redifined the parametrization on surface?

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 ...
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vote
2answers
60 views

Diffeomorphism ( differential geometry) [closed]

What is the Geometrical interpretation of diffeomorphism in context of differential geometry ?
0
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0answers
18 views

equivalence relation between two basis in a vector space

having the same orientation is an equivalence relation, for the different basis in a vector space. What is the purpose of equivalence relation in the sentence above?
0
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1answer
42 views

Easy example of unit speed plane curve?

I was trying to find a non-trivial example of a unit speed plane curve. The reason is I want something to work with but if I start with a non-unit speed curve and then do the arc length ...
4
votes
1answer
55 views

Calculation of $A'(0)$ (first variation of the area functional).

I'm trying to do the calculation that shows that a surface in $\Bbb R^3$ is area minimizing if and only if the mean curvature is zero. I'm getting a sign wrong and I'm going crazy, I need help. ...