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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Question for experts in dynamical systems or symplectic geometry

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof): In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...
4
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25 views

Complete Riemannian metric on ${\mathbb R}^2\setminus\{0\}$.

It seems to me that the Riemannian metric $g_{ij}=\delta_{ij}/|x|^2$ on the punctured plane is complete, but I don't find a proof not involving explicit computations of the geodesic equation. Does ...
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10 views

find the tangent space of hyberboloid?

How can I find the tangent space of the hyberboloid $$ x^2 +y^2 -z^2=a$$ for $$a>0$$ in the given point: $$(\sqrt{a},0,0)$$?
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1answer
34 views

Intuitively, what is the difference between homeomorphism and diffeomorphism? Significance?

As the title suggests, intuitively, what is the difference between homeomorphism and diffeomorphism? Many thanks in advance. What is the significance of such a difference?
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22 views

Taylor series representation for a Riemannian hypersurface

This is Exercise 8.5 in Lee's Riemannian Manifolds: An Introduction to Curvature. Suppose $M\subset \mathbb R^{n+1}$ is a hypersurface with the induced metric. Let $p\in M$, and let ...
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1answer
54 views

Evaluate integral with gaussian curvature

I thought evaluating it in the following way: $$\begin{align} \int_0^{2\pi}\int_0^{\pi}K(x,y)\sqrt{\det(g_{ij})} \, dy\,dx &= \int_0^{2\pi}\int_0^\pi \sqrt{\det L_{ij}}\cdot \sqrt{{\frac{\det ...
3
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2answers
47 views

Definition of a complex fiber

We define a real hypersurface as a subset $M\subset\Bbb C^n$ which is locally defined as the zero-locus of some $r\in\mathcal C^2(\Omega,\Bbb R)$ ($\Omega\subseteq\Bbb C^n$ open). Then let $z_0\in M$. ...
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22 views

Implicitization of Parametric Curves

I've got a 3D parametric, smooth, simple, and closed curve given by $\sigma(s) = (\sigma_1(s),\sigma_2(s),\sigma_3(s))$ where $\sigma_1(s)$ and $\sigma_2(s)$ are given by trigonometric functions of ...
2
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2answers
25 views

Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
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1answer
14 views

Tangent space of hyperboloid at a point [on hold]

How can I find the tangent space of the hyperboloid $x^2 +y^2 -z^2 =a$ for $a\gt 0$ at the point $(\sqrt{a},0,0)$?
5
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42 views

Higher-Order Differential Operators as Vector Fields

On a $C^\infty$ manifold $M$, one can produce the tangent space $T_p M$ at a point by equivalence classes of tangents to smooth curves through the point $p$. When realised this way, the tangent ...
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1answer
30 views

Diffeomorphism between Euclidean space

How does one show that if $f:U\rightarrow V$ is a diffeomorphism between open sets $U\subset\mathbb{R}^m$ and $V\subset\mathbb{R}^n$ then $m=n$? Here is some working: For $u\in U$ let $v=f(u)\in V$. ...
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0answers
30 views

Length vs area on a surface (tangent v. not in the plane, why ok for area and not for legth)

Tangent vectors on a curved surface do not lie in the plane, which is why, we cannot compute the length between two points, as $\sqrt{dx^2+dy^2}$. Yet, we are able to compute the area, by integrating ...
4
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0answers
47 views

Normal curvature of geodesic spheres

I would like to ask the community for a reference on the following property of geodesic spheres. Let $(M,g)$ be a compact Riemannian manifold without conjugate points and $\tilde{M}$ its universal ...
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0answers
44 views

Natural operators in differential geometry?

Which operators in differential geometry is called natural? And this neutrality is respect to what property or structure? Why this is an important problem? and what is due problem relations with lie ...
2
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0answers
39 views

A Quotient of the Euclidean Group

$\newcommand{\euc}{\mathscr I}\newcommand{\R}{\mathbf R}$ Let $\euc(n)$ denote the the Euclidean group $\R^n\rtimes O_n(\R)$. Recall that $\euc(n)$ acts on $\R^n$ as $(\mathbf x, T)\cdot \mathbf ...
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0answers
52 views

Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...
2
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1answer
79 views

Difficult to read about different subjects simultaneously, should I leave one for now? [on hold]

I learn math by reading books. Usually I read 3 books (about 3 different subjects) simultaneously and switch focus every couple of days. The books i'm studying right now are Rudin's functional ...
4
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1answer
26 views

Proving that the coordinate basis is a basis of a tangent space

Given a differentiable manifold $M$ and some chart $(U, \psi)$ near $p$, we can consider the curve $\tilde{\beta}_i: t \mapsto \psi(p)+t e_i$, where $e_i$ denoted the standard basis in $\mathbb{R}^n$, ...
2
votes
2answers
22 views

Closest point of parameterized curve has orthogonal position vector to tangent

Let $\alpha(t)$ be a parameterized curve which does not pass through the origin. If $\alpha(t_0)$ is a point of the trace of $\alpha$ closest to the origin and $\alpha'(t_0)\ne 0$, show that the ...
2
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1answer
49 views

Derivations of important algebras?

After knowing the importance of studying derivations of an algebras(Why we wonder to know all derivations of an algebra?), this problem naturally raised "what is the space of all derivations of ...
3
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37 views

looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
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1answer
21 views

Prove that regular curves are locally invertible

Consider the function $F = (F_1, F_2)$ from $I = (a, b) \subset \mathbb{R}$ to $\mathbb{R}^n$ (without loss of generality, assume $n = 2$). Suppose $F$ is differentiable (i.e $F_1' = f_1$ and $F_2' = ...
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1answer
37 views

Can the twisting of mobius band be represented by a U (1) bundle?

With the usual embedding of a mobius band, the strip is twisted by an angle pi, smoothly, as it goes round.I think this can be represented intrinsically, independent of the embedding, by attaching a ...
3
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1answer
42 views

Involutive distributions?

How do we check exactly that a distribution is involutive? I have the following definition in my book: A $k-$dimensional distribution $\Delta$ on a manifold $M$ is a smooth choice of a k-dimensional ...
7
votes
1answer
116 views

Why we wonder to know all derivations of an algebra?

It is well-known that the space of all derivations of algebra of smooth functions on a manifold is its space of sections (vector fields on underlying manifold). But, I wonder to know why finding the ...
4
votes
2answers
106 views

Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
2
votes
2answers
44 views

Under what conditions can a general 2-form be written as a wedge product of two 1-form

Assume we have a 2-form $\omega \in \Lambda^2\mathbb{R}^n$. It is usually stated one can write $$\omega = \alpha \wedge \beta,$$ with $\alpha, \beta \in \Lambda^1\mathbb{R}^n$ only for $n < 4$. How ...
4
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0answers
57 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
7
votes
1answer
110 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
2
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1answer
65 views

Convexity under diffeomorphisms

Let $K \subset \mathbb{R}^n$ be a compact convex subset with non-empty interior, and $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism. Then is it true that $f[K]$ is convex?
2
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1answer
47 views

Norm of clifford operator on forms

I am a beginner in differential geometry and would appreciate some pointers on how to answer the following question. Let $M$ be a closed orientable Riemannian manifold with $\{e^1,...,e^n\}$ an ...
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0answers
18 views

Representation of conjugate directions

Is there a way to represent conjugate directions on a Mohr circle of curvature? ( Surface Theory, Second fundamental form, M = 0 ) Directions given by double angles AOB, AOC. Is this attempt ...
3
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0answers
38 views

Holonomy computation in $S^2$

If $\gamma$ is a closed Loop in $S^2$ and $p\in S^2$, where $\gamma$ is the boundary curve of some region $X$ in $S^2$ (and $\gamma$ satisfied some regularity conditions), someone told me that the ...
2
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0answers
34 views

Spinors and forms

In this link http://benasque.org/2009gph/talks_contr/074Herdeiro.pdf page 15, it was said that: "Use spinorial geometry techniques: One takes the space of Dirac spinors to be the space of ...
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1answer
17 views

Countable intersection of Cut Locuses is always empty?

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then does there always exist a countable collection of points $\{p_n\}_{m \in \mathbb{N}}$ such that: \begin{equation} ...
3
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1answer
59 views

Is Heisenberg group Euclidean?

I'm reading an article speaking about Heisenberg group $\mathbb H^n$ and some of its properties. Now, I have some questions to ask, hoping to be clear enought. Reading the introduction I've ...
4
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1answer
69 views

Are lightlike curves in the De Sitter space straight lines?

I think that every lightlike curve in $\mathbb{S}_1^2 \subseteq \mathbb{L}^3$ must be a line. But I'm having trouble concluding it. Let $\alpha\colon I \subseteq \Bbb R \to \Bbb S^2_1 \subseteq \Bbb ...
5
votes
2answers
87 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
2
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1answer
81 views

Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
3
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0answers
21 views

If $N$ is the boundary of Riemannian $M$, can I compute $i^{*}(* (\alpha \wedge \beta))$?

There wasn't enough room in the title to explain completely: $M$ is an oriented Riemannian manifold with boundary $N$. $\alpha$ and $\beta$ are differential forms on $M$, $*$ denotes the Hodge star, ...
4
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1answer
40 views

Classical differential operators with complex functions on Riemannian manifolds

I am having some trouble understanding how to use the classical operators ($\nabla, \operatorname{div}, \Delta$) with complex functions on a Riemannian manifold $(M, g)$. Consider the formula ...
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0answers
25 views

2-dimensional Riemann Manifold

I am looking for a proof of the theorem that states that any 2-dimensional Riemann Manifold is conformally flat in the case of a metric of signature 0, following through with Problem 6.30 in the text ...
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0answers
48 views

“Maximum point lies on a curve” implies tangential derivative is zero there.

Given a differentiable function $f:\mathbb{R}^2\to\mathbb{R}$, suppose that it has a local maximum at the point $(x_0,y_0)$. Let $\gamma$ be a smooth curve passing through $(x_0,y_0)$. Does it follow ...
2
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1answer
80 views

Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
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1answer
24 views

Characterizing a surface

can somebody help me get started with this problem? I don't even know how to start the proof. Say $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable. Prove that $z=xf(y/x)$ belongs to a surface ...
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34 views

Computation in Wikipedia's article “Riemann Curvature Tensor”

This Wikipedia article explains how the Riemann curvature tensor is a measure of the failure for a tangent vector to parallel translate back to itself along an infinitesimally small loop. The article ...
2
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1answer
30 views

Understanding Symmetric tensor field

I am reading an article in which author calls some basic tensor analysis result. He states in general we define on $\mathbb R^N$ that $$ \mathcal T^k(\mathbb R^N):=\{\xi:\,\mathbb ...
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52 views

Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
3
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1answer
51 views

Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...