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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Riemann tensor symmetries

The Riemann tensor has its component expression: ...
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Two questions about Li-Yau-Hamilton estimate

Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I can't see $Q\ge 0$ when $t=0$. Besides, how to ...
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30 views

Gaussian Curvature of a Pseudosphere.

I have been trying to find the gaussian curvature of a pseudo sphere. I assumed the parametrization: X(u,v) = (cos(u)*sech(v), sin(u)*sech(v), u - tanh(u)). I know that it's a surface of revolution ...
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Whether the tangent space can be saw as $M\times R^m$?

Let $M$ be a smooth differential n-dim manifold, $TM$ is the tangent space ,I think the $TM$ can be treated as $M\times R^m$ ,whether it is right ?
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Linear connection on a 1-form

Let $M$ be a manifold with linear connection $\nabla$ and let $X$ be a vector field on $M$. Given a 1-form $\alpha \in \Omega^{1}(M)$, define $\nabla_{X} \alpha : \scr{X}$ $(M) \to C^{\infty}(M)$ by ...
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22 views

Support Vector machine & Support Vector

I had gone through several example of SVM and I see one starts explaining SVM by picking up the support vectors upfront (like this https://www.youtube.com/watch?v=1NxnPkZM9bc). Basically those vectors ...
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1answer
40 views

Arc-length parametrization

Let $\gamma:[a,b]\times \mathbb R\to\mathbb R$ be a flow of plane curves given by $$\dot\gamma=\frac{\kappa'}{\vert\gamma'\vert}JT+\frac{1}{2}\kappa^2T$$ where $T$ is the unit tangent vector and ...
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1answer
203 views

What does it mean for the group of rotation matrices to have a “manifold structure”

I have just been exposed to rotation matrices, and it is showing extremely strong connection with group theory and differential geometry which I am both totally inept at. Can someone in simple term ...
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21 views

Immersions-possible error in Dieudonné III?

Below I refer to [D] Dieudonné Treatise on analysis III [B] Bourbaki VARIETES DIFFÉRENTIELLES ET ANALYTIQUES [M] Michor Topics in differential geometry In [D,16.7.7], we can read: Let $f ...
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24 views

Definition of a foliation

What are the relations among the several submanifolds of a foliation, if any? Each submanifold has some atlas, is there any relation, by definition, among the several atlases?
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Srednicki's QFT - chapter 2 - understanding from a mathematician's point of view

I am reading the first chapters of Srednicki's Quantum Field Theory book, trying to understand them from a mathematician's point of view. In particular, I'm interested to what happens when you try to ...
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30 views

On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
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52 views

Proof of theorema egregium of Gauss

My problem is to solve this exercise : Let $S \subset \mathbb{R}^3$ be a sybmanifold of dimension $2$, and $x:U \mapsto S$ a local parametrization at $p = x(u,v) \in \mathcal{S}$. We have (admitted) ...
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1answer
34 views

Prove there exists a smooth unit normal at the boundary of the following manifold

Let $M$ be a compact subset of $\mathbb{R}^3$ with smooth boundary $S=\partial M$. Consider M with the standard orientation $\mu=\mu_{0}$ from $\mathbb{R}^3$ and $S$ with the boundary orientation ...
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16 views

Principal bundle isomorphism.

Let $G\longrightarrow P\overset{\pi}{\longrightarrow} M$ be a differentiable principal bundle, i.e. $M$ and $P$ are differentiable manifolds, $G$ is a Lie group, $\pi$ is a differentiable surjective ...
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20 views

Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
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49 views

Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
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1answer
340 views

Klein Bottle Embedding on $\mathbb{R}^4$.

First of all, I am aware of the question in How to embed Klein Bottle into $R^4$ , which was inconclusive. Anyway, I've made some progress, but I still have a question. I am using Do Carmo's ...
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1answer
49 views

Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
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1answer
24 views

What curves have a closed-form formula for projecting a point onto them in multiple dimensions?

What curves have a closed-form formula for projecting a point onto them in multiple dimensions? For example, give a simple, straight line $$ c(t) = v t $$ where $v\in\mathbb{R}^m$ and ...
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1answer
66 views

How to determine A?

I'm struggling with the following problem: Let $(\gamma, \tau)$ be an arc-length parametrized curve and $\mathcal A: \mathbb R^2 \to \mathbb R^2$ be a Euclidean transformation so that the curve $\hat ...
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1answer
66 views

How do I determine a cricital point of an area functional?

The orientated area $A(\gamma)$ of a regular closed plane curve $(\gamma, \tau)$ is defined as $$A(\gamma) :=\frac{1}{2}\int_{0}^\tau \det (\gamma,\gamma')$$ Now how can I determine the cricital ...
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2answers
47 views

Calculating tangent space of $x^{2}+y^{2}=z^{2}$ at origin

I am asked to show that the tangent space of $M$={ $(x,y,z)\in \mathbb{R}^3 : x^{2}+y^{2}=z^{2}$} at the point p=(0,0,0) is equal to $M$ itself. I have that $f(x,y,z)=x^{2}+y^{2}-z^{2}$ but as i ...
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17 views

For manifolds $M,N$ show that $W^{1,p}(M,N)$ is path-connected iff $C^0(M,N)$ ist path-connected.

I'm asked to show that for compact, smooth Riemmanian manifolds $M,N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from ...
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1answer
42 views

Elastic curves - What is wrong about my solution?

Given a curve $\gamma:\mathbb R\to\mathbb R^2$ with $\Vert\gamma'\Vert=1$ and curvature $\kappa(s)=\frac{c}{\cosh s}$, $c\in\mathbb R$, how can I show that $\gamma$ is an elastic curve for some ...
2
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1answer
30 views

Showing that the Hopf fibration is a non-trivial fibre bundle

I want to show that the Hopf bundle $$ \mathbb{S}^1 \rightarrow \mathbb{S^3} \rightarrow \mathbb{S}^2$$ is non-trivial as a principal fibre bundle. I have seen hints of several different approaches: ...
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1answer
20 views

Degree of smooth map of manifolds depends on orientation choice?

I'm a little to confused as to why it appears that the degree of a smooth map $f: M \to N$ between smooth manifolds appears to only be defined up to sign - I'm not sure where my mistake is. By ...
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1answer
28 views

push forward of the levi civita connection

Let $M$, $M'$ be riemann manifolds with levi-civita connection $\nabla$,$\nabla'$. If $\phi$ is an isometry (global so diffeomorphism too) I want to show: $ \nabla'_{X'} Y'=D\phi (\nabla_X Y) $ where ...
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1answer
29 views

Nilpotent Lie subalgebra of Lie algebra of Killing vector fields

Suppose $M$ is a smooth manifold with Riemannian metric $g$. Recently I have dealt with some problem which lead me to the following question: Can a Lie algebra of Killing vector fields on $M$ has a ...
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48 views

Intuitive, short explanation of differential forms and exterior calculus

Are there any introductory lecture notes on differential forms and exterior calculus, preferably aimed at physics students studying General Relativity and Black holes? I have some familiarity with GR ...
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geodesic flow is proper action

Good evening to everyone. I'm having a problem in the following setting: If I'm having a homogeneous manifold $M=G/K$, where $K \subset G$ is a closed subgroup, I can always find a $G$-invariant ...
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101 views

Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E*(du^2+dv^2)$ and where $E>0$ is an harmonic function? I know that this metric is a special kind ...
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3answers
360 views

Möbius strip as a non-trivial principal bundle

There is a well-known theorem that a principal bundle is trivial if and only if it admits a global section. I'm trying to get a good picture of what this theorem means. The Möbius Strip can be ...
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Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
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A question on diffeomorphisms of a manifold

we know that any vector field $X$ on a smooth manifold $M$ generated the 1-parameter group $\phi_t$ ( the locally diffeomorphisms). My question: Are the locally diffeomorphisms on the manifold $M$ ...
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Minimal requirements to be a submersion.

I saw here (A surjective map which is not a submersion) that a smooth differentiable map $f:M\to N$ between two manifolds $M$ and $N$ is not necessarily a submersion. A counterexample is ...
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Showing that Killing vector fields form a vector space without introducing connection

I'm trying to show that the sum of two killing vector fields is again a vector field without introducing a connection and just using the smooth structure on a manifold. Let $X,Y$ be Killing vector ...
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3answers
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degree 1 map $f : M \to S^n$

Let us consider $M$ a closed and connected n-manifold which is also orientable. An exercise in Hatcher claims that for any such $M$ there is a continuous map $f: M \to S^n$ such that it's degree is 1, ...
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Progressively embed ( = superscribe?) and immerse …

$\mathbb R^1$ is superscribed/embedded on $\mathbb R^2$ and $\mathbb R^2$ in turn immersed in $\mathbb R 3$. Graph of a line $ x(u,v), y(u,v), z(u,v),f(u,v)=0 $ is superscribed or embedded on ...
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1answer
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About the parallel transport

Definition 1: Let $M$ be a differentiable manifold with an affine connection $\nabla$. A vector field along a curve $c:I\to V$ is called parallel when $\dfrac{DV}{dt}=0$ for every $t\in I$. ...
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4answers
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Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that I have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
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3answers
62 views

Showing de Rham cohomology $H^1(S^n)$ is zero

I'm trying to find an elementary way to see that the 1st de Rham cohomology of the n-sphere is zero for $n>1$, $H^1(S^n) = 0$. This is part of an attempt to find the de Rham cohomology of the n ...
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2answers
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Trying to prove that $TM$ is a manifold: Is this function an homeomorphism?

I am trying to prove that if $M$ is a $k$-manifold in $\mathbb R^n$, then $TM=\{(p, v): p \in M, v \in T_pM\}$ is a manifold. Here, $T_pM$ is defined as a subset of $\mathbb R^n$. I know that ...
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1answer
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Defining a contact form

I am trying to understand contact structures. To this end, as an exercise, I intend to define a contact form on $S^3$. Here is what I have so far: Since $S^3$ is in $\mathbb R^4$ one can specify a ...
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Why do people care about principal bundles?

I've started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought ...
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1answer
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Why is a diffeomorphism preserving a parallelism locally uniquely determined by its value at $1$ point?

According to the title, why is a diffeomorphism preserving a parallelism locally uniquely determined by its value at $1$ point?
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Integral curves of time dependent derivations

Question: Given smooth manifold $M$, with algebra of smooth functions deoted by $C(M)$ let $D_t$ be a time-dependent derivation of $C(M).$ Let $\hat{D}$ be a derivation of $C(M\times \mathbb{R})$ ...
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Connection planes associated to differential 1-forms

In textbook in differential geometry such an idea appears: there is a tangle bundle $\pi:TM\rightarrow M$ and we are actually looking at the trivialization $\pi^{-1}(U)=U\times \mathbb{R}^n$. We are ...
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Every real matrix with non-negative entries has a non negative eigenvalue [duplicate]

If $A$ is any matrix $n\times n$ with non negative entries, then $A$ has a non negative eigenvalue. I know that I have to use the Brower Point fix theorem, but I am not finding the function for that. ...