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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Conditions on characteristic polynomial to define a matrix submanifold.

I'm trying to find conditions on the characteristic polynomial, $p$, of a matrix such that the pre-image of matrices with characteristic polynomial $p$ form a manifold. More precisely, we can write ...
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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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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 ...
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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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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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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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32 views

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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1answer
48 views

Maps between tangent space of product manifold and sum of tangent spaces

I am trying to prove that $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$ We define: $$\Phi:T_{(p,q)}(M\times N)\to T_pM\oplus T_qN:v\mapsto(d_{(p,q)}\pi_M v,d_{(p,q)}\pi_N v)$$ and $$\Psi:T_pM\oplus ...
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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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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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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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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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2answers
28 views

Local defining functions on real hypersurfaces.

I'm currently reading Real Submanifolds in Complex Space and their Mappings by Baouedi, Ebenfelt and Rothschild. I'm currently stumped by what the author's claim to be an easy check (really blowing ...
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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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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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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 = ...
0
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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 ...
2
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1answer
33 views

How can I prove that interior product obeys a graded Leibniz rule?

I want to prove that $i_{X}(\omega\wedge\phi)=i_{X}\omega\wedge\phi+(-1)^{k}\omega\wedge i_{X}\phi.$ I was thinking I many be able to adapt the proof that the exterior derivative obeys the graded ...
23
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498 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
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2answers
43 views

Injective curve

How can i show that the curve $\gamma:\mathbb{R}\to\mathbb{R}^2$ defined by \begin{equation} \gamma(t)=\left(\frac{t}{1+t^2},\frac{t}{1+t^4}\right) \end{equation} is injective (without using algebraic ...
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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: ...
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14 views

Are the coordinates of geodesic curves in an analytic manifold, analytic functions?

I wonder if the coordinates of geodesic curves in an analytic manifold, analytic functions? Thanks in advance.
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1answer
35 views

Is a geodesic always a rectifiable curve?

I am not an expert in differential geometry, but I need to know the following If any geodesic that joins two points in a compact and Riemannian manifold is necessary a rectifiable curve, or there ...
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1answer
45 views

Arc length of implicit curve using gradient magnitude of the unit step function?

I came across a funny formula for the arc length of an implicit curve in the paper here, given at the top of page 5. Let me set the context: Consider a function $\phi: \mathbb{R}^2 \to \mathbb{R}$ ...
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18 views

Euler characteristic of branch cover of punctured Riemann surface

Let $\Sigma_1$, $\Sigma_2$ be two closed Riemann surfaces, $\pi: \Sigma_1 \to \Sigma_2$ is degree $m$ branched cover of $\Sigma_2$, then we have formula about their Euler number: $$\chi(\Sigma_1)= ...
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2answers
41 views

Invariant forms on a manifold

Probably a silly question - still, it's been bugging me for some time now. Say that we have an invariant $1-$form $\omega$ on a smooth manifold $M$, acted on by a group $G$. Then \begin{equation} ...
3
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2answers
158 views

Homotopy Poincaré conjecture - no map inducing the isomorphism on homology

$\newcommand{\Z}{\mathbb{Z}}$ In Terence Tao's notes on page 18, concerning the Poincaré conjecture, he gave the following sketchy proof of the homotopy Poincaré conjecture. Given $M^3$ a 3-manifold ...
3
votes
1answer
450 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
4
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1answer
42 views

Showing that $\mathbb{R}$ is locally isometric to $S^1$

Show that $f:\mathbb{R}\to S^1$ given by $f(t)=e^{i t}$ is a local isometry between Riemanninan manifolds. So, basically we need to show that for each $p\in\mathbb{R}$ there exists ...
0
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1answer
45 views

Consider the function : $L_{ij}=x_{i}\frac{\partial}{\partial x_j}-x_{j}\frac{\partial}{\partial x_i}$

Let $\Omega$ be a smooth bounded domain of $\mathbb{S}^n$, the unit $n$ sphere centered at the origin of $\mathbb{R}^{n+1}$, and consider the functions $$L_{ij}u=x_{i}\frac{\partial u}{\partial ...
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1answer
49 views

Coordinate charts vs. coordinates on manifolds

I just realised that I'm confused what coordinates really means in the context of manifolds. For example, say $M=S^2$. Then we can define smooth charts as follows: Let the open sets be $U = S^2$ ...
3
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58 views

What are the essential tools and proof techniques for beginning smooth manifolds and differential topology?

I am an undergraduate currently taking a first course in smooth manifolds. I feel that I understand the material intuitively. But, I'm having trouble turning my intuition into proofs. I was hoping ...
2
votes
1answer
56 views

Thoughts on Theorema Egregium

Due to this theory any mapping from the globe to a paper neccesairly have disortions. My question is there a theory which states the number of necessary map that gets this error down to a certain ...
2
votes
1answer
39 views

Can we find a regular ($C^k$) parametrization for this surface?

I have here a surface whose curvature properties I want to study, represented in cylindrical coordinates: $$f(r,\theta) = r^2\cos4\theta$$ The problem, however, is that the parametrization is not ...
2
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1answer
16 views

How do you compute a complex exterior derivative?

The context is deriving cauchy riemann equations using green's/stoke's theorem. The function is the complex function $f(x,y)=u(x,y)+iv(x,y)$ with associated one form $u(x,y)dx+iv(x,y)dy$. Here is my ...
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2answers
30 views

Given $p \in S^{n-1}$, how does one show that the map $\pi: SO_n \to S^{n-1}$ by $A \to A(p)$ is a submersion?

Pick $ p \in S^{n-1} \subset \mathbb{R}^n$ and consider the map $\pi: SO_n \to S^{n-1}$ by $A \to A(p)$. Show that this map is a smooth submersion. For $ q \in S^{n-1}$, describe the pre-image. For ...
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25 views

A necessary condition for a smooth map to be a local isometry - why should the integrands be equal?

This is theorem $6.2.2$ - page $127$ of Pressley's Elementary Differential Geometry: A smooth map $f:S_1 \subset \Bbb R^3 \to S_2 \subset \Bbb R^3$ is a local isometry if and only if the symmetric ...
3
votes
1answer
32 views

How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
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23 views

On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?

Let $M$ be a parallelizable manifold. Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$ for all $i,j$ ? If the answer is no, what kind of obstruction there is to find such a frame ? ...
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Find a graduate/research-level math tutor [closed]

I'm coming from the social sciences and decided about 5 years ago to research the topic of artificial intelligence. Having mastered the basics, I would now like to design some experiments of my own, ...
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1answer
22 views

Does parallelizability by a frame of commuting vector fields imply those fields are coordinate vector fields of a global chart?

If $M$ is a parallelizable manifold, is the following true? If $(X_i)$ is a (global) frame of $M$ and $[X_i,X_j]=0$ for all $i,j$, then there is a global chart for which $X_i=\partial_i$.
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1answer
31 views

Why is the 'line-element' non-integrable?

I'm reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt: A second landmark is the geometry of Riemann, which grew out of the ingenious ...
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1answer
29 views

Such a manifold is homeomorphic to a sphere

I recently read that if a compact differentiable manifold admits a real function with only two critical points, then it is homeomorphic to a sphere. If the function is Morse, this follows from ...
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1answer
21 views

Choice of a horizontal tangent space of a principal bundle

Let $\pi:P\to M$ be a principal bundle with group $G=\pi^{-1}(p)$, and let $u\in P$ and $p=\pi(u)$. As I understand it, the choice of the vertical tangent space $V_uP=\mathrm{ker}(\pi_*)$ is natural, ...
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20 views

Area of a complete, simply-connected surface with non-positive Gauss curvature is infinite

I am reading an article of F. Xavier about the Gauss map of complete, non-flat minimal surfaces in $\mathbb{R}^3$ (reference: http://www.jstor.org/stable/1971139?seq=1#page_scan_tab_contents). In the ...
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1answer
54 views

Gradient and Divergence in Riemannian Manifold

Let $M$ a riemannian manifold. Let $X\in\chi(M)$ and $f$ a function $C^{\infty}$ in $M$. Define the divergence of $X$ as a function $div X:M\to\mathbb{R}$ given by $divX(p)=\mbox{trace of the linear ...
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1answer
19 views

Invariance of the rank of the trace of Riemannian curvature under a change of frame

Let $$ R=\begin{pmatrix} R_{11} & ... & R_{1n} \\ &...\\ R_{n1} &...& R_{nn} \end{pmatrix}, $$ where $R_{ikjl}$ is curvature tensor of a Riemannian manifold $(M,g)$ and ...
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31 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
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1answer
25 views

Gauss application is an isometry

My question is very simple : What can we say of a compact surface $\mathcal{S} \subset \mathbb{R}^3$ satisfying that the Gauss application $N$ is an isometry : $\mathcal{S} \to \mathbb{S}^2$ ...
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44 views

Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ ...