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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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 ...
2
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1answer
38 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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2answers
43 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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16 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 ...
2
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1answer
27 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: ...
1
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1answer
58 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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1answer
15 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 ...
0
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1answer
41 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 ...
0
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1answer
31 views

Is there any difference between Immersion and embedding?

Definition as below , I think they are same ,is right ?
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1answer
28 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 ...
0
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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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45 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 ...
0
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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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0answers
29 views

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 ...
2
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0answers
83 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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24 views

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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0answers
25 views

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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0answers
24 views

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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0answers
12 views

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 ...
0
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1answer
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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3answers
48 views

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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1answer
32 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 ...
0
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1answer
30 views

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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1answer
43 views

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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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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19 views

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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0answers
22 views

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. ...
3
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1answer
36 views

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?
5
votes
1answer
55 views

Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
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3answers
55 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
32 views

Maps between manifolds with boundary and homeomorphism

Assume we have $f:(M,\partial M)\rightarrow (N,\partial N)$ connected 3-manifolds, not compact, such that $f$ is an homeomorphism onto its image and $f(\partial M)=\partial N$. Can say that $f$ has to ...
0
votes
1answer
47 views

Conformal map is an isometry

I have the upper half-plane $\mathbb H$ with the metric given by $$\mathrm ds^2=\frac{1}{y^2} (\mathrm dx^2+\mathrm dy^2)$$ and the unit disk $\mathbb D$ with the metric given by $$\mathrm ...
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0answers
10 views

Local isometries preserve director curve

I'm reading about local isometries of ruled surfaces. Ruled surface is parametrized by $f(u,v)=c(u)+ve(u), v,u \in R.$ Curve $c$ is called the base curve, and curve $e$ is director curve. $v$-curves ...
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1answer
135 views
+50

How can we define $\partial x_{i_r}^p(X_p^r)$?

Suppose $M$ is a manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ and $\mathcal ...
2
votes
0answers
30 views

Lagrangian densities, Lie Groups and Lie Algebras

I'm quite new to Physics and I was having a look for the first time to the Standard model. I'm not sure if the mechanism that I'm describing is directly from Weyl or from others but what I found quite ...
3
votes
2answers
90 views

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 ...
0
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0answers
17 views

Determining a normalization for a function on the three-sphere

I'm trying to find a normalization condition for $\Phi$ for the following problem on $S^{3}$ (note that this is NOT the unit three-sphere but has a radius R: $$\oint_{\partial ...
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A exercise of Riemannian geometry.

Let $(V,g)$ be Euclid vector space , $a$ is a symmetric 2-tensor , define $a^*: V\rightarrow V$ as $$ \langle a^*(X) , Y \rangle =a(X,Y) ~, ~~~~~~ X,Y\in V $$ $\langle~,~\rangle$ is inner product . ...
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0answers
33 views

Defining a differential for quotients

Let $f \colon M \to N$ be a smooth map between smooth manifolds and $f$ being a surjective submersion. Assuming we have a proper Lie-group action $G$ on $M$, with only one orbit type and $G$ acts on ...
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2answers
45 views

Killing form is negative definite

Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex ...
2
votes
2answers
59 views

Is topology invariant under conformal transformation?

Can conformal transformation change the topology of a manifold? In other words, if two manifolds are conformal, should they have the same topology?
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0answers
18 views

Parametrisation of a differential equation by arc length

Suppose I have the following PDE: $\frac{\partial y}{\partial t} = \frac{\partial}{\partial x}\left[(1-y)^3y^3\left(\sin \theta + \frac{\partial y}{\partial x}\cos \theta\right) \right]$ I notice ...
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+50

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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0answers
45 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as ...
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2answers
83 views

$\operatorname{SU}(n)$ as manifold

I am trying to do this has a while, but I cannot use correctly the regular value theorem to do so! I appreciate any help. The problem is that I cannot choose the function to take $SU_n$ as a regular ...
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0answers
27 views

Mobius strip as manifold and as a bundle over $S^1$

I am trying to construct the Mobius strip bundle onver $S^1$. I am stuck in how to define charts for the Mobius bundle. Once I make this, the problem will easily be solved. My attempt was: $$M = ...
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0answers
30 views

Smooth coverings are open maps proof verification

Let $M,N$ be connected, smooth manifolds. A function $F:M \rightarrow N$ is a local diffeomorphism if for all $p \in M$ there exists open $U \subseteq M$ with $p \in U$ such that $F(U) \subseteq N$ is ...
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1answer
36 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on ...
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1answer
39 views

Area of circle in terms of Gaussian curvature

I am asking about a formula in section 2 of these notes. Let $\rho|dz|$ be a conformal metric on $U\subset\mathbf C$. Then the Gaussian curvature of $\rho|dz|$ at $z\in U$ is defined as ...
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12 views

$S_{2}(f)=0$, with $f$ nonconstant. Applications of the Hessian operator.

Study article R. C. Reilly is entitled Applications of Hessian operator in the Riemann manifold had a doubt in the remark, shortly after the theorem 2 of that Article. The theorem is stated as: ...