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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Prove that $X$ parametrizes a regular surface $M$ in $\mathbb{R}^3$ and determine for which values $p$ the curve $y$ is geodesic on $M$.

Real valued functions $f,g:\mathbb{R}_+ \rightarrow \mathbb{R}$ are $f(u) = e^{-u}$ and $g(u) = \int \sqrt{(1-e^{-2t}} dt$. Given $X:\mathbb{R}_+ \rightarrow \mathbb{R}^2$ with $X(u,v) = [f(u) \cos ...
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128 views

Does diffeology provide moduli for classical constructions?

Do classical constructions on differentiable manifolds like affine connections, Riemannian metrics, or (almost) complex structures have moduli spaces in category of diffeological spaces?
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1answer
49 views

is a non-falling rank of smooth maps an open condition?

If $f \colon M \to N$ is a smooth map of smooth manifolds, for any point $p \in M$, is there an open neighbourhood $V$ of $p$ such that $\forall q \in V \colon \mathrm{rnk}_q (f) \geq \mathrm{rnk}_p ...
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67 views

Complex structure on the product of two complex Kähler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kähler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
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37 views

Why this map is identically $0$?

I encountered following proposition: Let $p$ be a prime, $R=\mathbb{Z}/p^2\mathbb{Z}$, $M=\mathbb{Z}/p\mathbb{Z}$ as $R$-module. Then we have the map $M^*\otimes_RM\rightarrow$Hom$_R(M,M)$ given by ...
3
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1answer
55 views

Linear map between $M^*\otimes_RN\rightarrow \text{Hom}_R(M,N)$

I'm reading a lecture note about tensors, following is a proposition: For $R$-module $M$ and $N$, there is a linear map $M^*\otimes_RN\rightarrow \text{Hom}_R(M,N)$ sending each elements ...
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1answer
519 views

Prove the regular surface with 2 geodesics from p to q, and negative curvature cannot be simply connected.

What ideas/formulas are required to solve this? Exercise: If a and b are two geodesics from point p to q, how do you prove that M is not simply connected? M is a regular surface in R3 and has negative ...
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1answer
103 views

Given a closed submanifold $Y$ of $X$ and a $C^{\infty}$ map $f$ on $Y$, can $f$ be extended to $X$?

I want to extend $f$ locally in the intersection of each coordinate patch of $X$ with $Y$, (and set it to $0$ outside of $Y$) and then use a partition of unity to get a differentiable map that agrees ...
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1answer
167 views

When does the normal vector of a Moebius-strip intersect?

In class the teacher was talking about normal vectors. $r = \langle x,y\rangle$ then the normal vector is $$ N\left(t\right) = \frac{T^{\prime}\left(t\right)}{||T^{\prime}\left(t\right)||} $$ where ...
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573 views

intrinsic proof that the grassmannian is a manifold

I was trying to prove that the grassmannian is a manifold without picking bases, is that possible? Here's what I've got, let's start from projective space. Take $V$ a vector space of dimension n, and ...
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245 views

Almost A Vector Bundle

I'm trying to get some intuition for vector bundles. Does anyone have good examples of constructions which are not vector bundles for some nontrivial reason. Ideally I want to test myself by seeing ...
12
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1answer
479 views

Differentiable manifolds as locally ringed spaces

Let $X$ be a differentiable manifold. Let $\mathcal{O}_X$ be the sheaf of $\mathcal{C}^\infty$ functions on $X$. Since every stalk of $\mathcal{O}_X$ is a local ring, $(X, \mathcal{O}_X)$ is a locally ...
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1answer
50 views

How can we find the arc length of the curve? [closed]

How can I find the length of the curve $$\left(\frac{t^3}{3} - t\right)\mathbf{i}+ t^2 \mathbf{j}, \quad 0≤t≤1?$$
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2answers
180 views

Explanation about frames as distinct from a co-ordinate system

I am quite confused as to what is the difference between a frame and a co-ordinate system. The wikipedia page was not very helpful for me. I would be very happy if someone could give me a non-rigorous ...
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71 views

Isomorphism between tensor product and set of all tensors

I'm new to tensor and quite confused. First of all, could anyone provide any friendly reference about tensors which explains the idea behind those boring-looking definition? Then is my problem. Let ...
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36 views

Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module?

As the title saying, Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module? Here we are considering vector bundles with finite-dimensional fibers.
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2answers
390 views

Hodge dual on orthonormal basis: two inconsistent answers

I'm trying to learn differential geometry using Göckeler & Schücker's book and I have some problems with the hodge star. As an example, say we have two orthonormal bases $e^i$ and ...
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1answer
248 views

How show that a surface embedded is non-orientable?

Let be $M$ a compact $3$-manifold. If $\Sigma$ is a embedded surface in $M$, such that $\Sigma$ is homeomorphic to $\mathbb{RP}^2$. If $i: \pi_1(\Sigma) \longrightarrow \pi_1(M)$ is not injective, ...
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1answer
74 views

Is any continuous curve in $\mathbb{R}^n$ a 1-D manifold?

I wonder if there is any theorem stating that any continuous curve in $\mathbb{R}^n$ is a 1-D manifold. If not, can anyone provide an example? At first I thought maybe a Peano curve affords a ...
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3answers
629 views

Example of non-manifold surface.

Is there any example of a surface which is locally homeomorphic to $R^n$ but is not a manifold? (i.e. does not have an well-defiend atlas)
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106 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
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2answers
152 views

Identifying functions on the unit disk with functions on the upper hemisphere

I've been wondering about something, and it might be nonsense (if so I apologize!). Consider the unit disk in $\mathbb{R}^2$ and a function $f$ defined on the disk. I can compute its double integral ...
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281 views

What is the intuition behind the Lie derivative of a vector field.

We have the following two formula about the Lie derivative of a vector field: $$ \left.\frac{d}{dt}\right|_{t=0}T\varphi_{-t}\cdot Y_{\varphi_t(p)}=[X,Y]_p = (\mathcal{L}_XY)(p) $$ where ...
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63 views

Smooth Monotone $\mathbb{R}^3$ curve with constant (nontrivial) curvature

So I was trying to construct a closed curve in $\mathbb{R}^3$ with constant positive curvature and non-trivial torsion. To do this I tried to glue two helices together in a smooth way with a curve ...
3
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1answer
373 views

Integral of a curve with respect to its curvature?

I've been struggling with this one for about $3$ weeks: What is the integral of a $\mathbb{R}^3$ curve with respect to its curvature? I though about approaching it with the Ferret-S formulas, and ...
3
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0answers
75 views

Vector field on torus as a submanifold of $\mathbb R^4$

Let $f(\theta,\phi)=\frac{1}{\sqrt{2}}(\cos \theta,\sin \theta,\cos \phi,\sin \phi)$ be immersion of torus into $\mathbb R^4$. How to prove that $\nabla_{\frac{\partial}{\partial \theta}} ...
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1answer
168 views

Transition functions on a quotient manifold

Here's an exercise given during a course in Differential Geometry that I'm taking. Let $M$ denote a smooth manifold and let $G$ be a finite group of diffeomorphisms acting on it without fixed points ...
4
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1answer
149 views

O(n) as embedded submanifold

I want to show that the set of orthogonal matrices, $O(n) = \{A \in M_{n \times n} | A^tA=Id\}$, is an embedded submanifold of the set of all $n \times n$ matrices $M_{n \times n}$. So far, I have ...
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1answer
112 views

Verification that $S^{n}$ is a differentiable manifold.

Setting $S^{n} := \{x\in\mathbb{R}^{n+1}: \|x\| = 1\}$, and labelling the north and south poles as $N:= (0,\ldots,0,1)$, $S:=(0,\ldots,0,-1)$, I can set the coordinate charts up as follows: Let $U_N ...
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353 views

Is the Empty set an orientable manifold?

The empty set can be regarded as an object in the category of smooth manifolds, at least for technical considerations. Is the empty set an orientable manifold?
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1answer
51 views

systole of space projective 3-dimensional

I'm study the papper "H. Bray, S. Brendle, M. Eichmair, and A. Neves, Area-minimizing projective planes in three-manifolds, Comm. Pure Appl. Math". (see http://arxiv.org/abs/0909.1665). Let be ...
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1answer
148 views

Question About Differential Functions Between Manifolds

Let $X$, $Y$ be manifolds, with respective coordinate charts $\{(U_{\alpha}, \varphi_{\alpha})\}_{\alpha\in I}$ and $\{(V_{\beta}, \psi_{\beta})\}_{\beta\in J}$. I want to show that $f:X\to Y$ is ...
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1answer
92 views

Composition of a differentiable map with a homemorphism.

Suppose I have a $C^{\infty}$ map $f:X\to\mathbb{R}^{n}$, for some differential manifold $X$. Then I also have a homeomorphic coordinate map $h$ from a subset of $\mathbb{R}^{n}$ to a subset of $X$. ...
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107 views

Identification of integration on smooth chains with ordinary integration

Let $M$ be a smooth oriented $n$-dimensional manifold and denote by $A \in H_n(M;\mathbb{Z})$ the fundamental class of $M$ (a generator of singular homology consistent with the orientation of $M$). ...
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426 views

$D^m\cup_h D^m$, joining $D^m \amalg D^m$ along the boundary $\partial D^m$

Given an orientation-preserving diffeomorphism $h: \partial D^m \to \partial D^m$, we can glue two copies of the closed unit disk $D^m$ along the boundary by identifying $x \sim h(x)$ to form the ...
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1answer
218 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
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Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange ...
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2answers
108 views

Tangent space on the north pole of $S^2$

We have 3 different (but equivalent) definition of tangent space, one of them is by equivalent class of smooth path, i.e. let $c:(-1,1)\rightarrow M$ be a path on $M$ with $c(0)=p$, then $c_1\sim c_2$ ...
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1answer
193 views

Conformal relation for 2-dim Lorentz space-times

I have two 2-dimensional space-times ($\mathbb{S}^1\times\mathbb{R}$) with signature $(-,+)$. One of them is flat the other one has non-vanishing curvature (Riemann tensor), both have vanishing Ricci ...
2
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1answer
139 views

Second Fundamental form in terms of defining function

I have an m-dimensional riemannian manifold M and an n-dimensional submanifold N that is given by $N = f^{-1}(0)$, where $f: M \longrightarrow \mathbb{R}^{m-n}$ ($0$ is supposed to be a regular value ...
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2answers
209 views

Circular Helicoid

A helicoid has the following parametric equation: $$ r(u,v)=\left \langle v\cos u,v\sin u,cu \right \rangle,\qquad u,v,c\in\mathbb{R}. $$ In ruled form, $$r(u,v)=\alpha(u)+v\Lambda(u),$$ it has ...
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1answer
175 views

First Integral of a Vector Field.

In do Carmo's Differential Geometry of Curves and Surfaces, In the Section about Vector Fields, first Lemma, he proves that for every differentiable vector field, there exists a function that is ...
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617 views

Any example of manifold without global trivialization of tangent bundle

It is said for most manifolds, there does not exist a global trivialization of the tangent bundle. I am not quite clear about it. The tangent bundle is defined as $$TM=\bigsqcup_{p\in M}T_PM$$ So is ...
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76 views

Problems about dual map, cotangent bundle.

I have no idea what dual space and dual map are, so have much trouble understanding cotangent bundle when reading Lee's book. First of all, can anyone give me a introduction what the dual map and ...
6
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1answer
531 views

Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
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1answer
54 views

Rank of the differential of a composition

Let $U$ be an open subset of $\mathbb{R}^2$ and $f:U\to \mathbb{R}$ a differentiable function. Let $S=\{(x,y,f(x,y)): x\in U\}$ be the graph of $f$. Let $V$ be an open subset of $\mathbb{R}^2$ and ...
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1answer
131 views

Group isomorphism for deck transformation in covering space.

When reading Lee's book, I encountered the following exercise: Let $\mathcal{P}\colon M\rightarrow G\backslash M$ be the covering arising from a free and proper discrete group action of $G$ on ...
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1answer
75 views

$Gr_2^+(\mathbb R^4) \cong S^2 \times S^2$

Let $Gr_2^+(\mathbb R^4)$ be the oriented Grassmanian of 2-planes in $\mathbb R^4$. How would one go about showing that this is diffeomorphic to $S^2 \times S^2$?
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120 views

Chain rules for differential forms

I have the variable $x,y,z$ possibly depending on each other i.e. on a smooth manifold. Using the theory of differential forms I can derive $\left(\frac{\partial x}{\partial y}\right)_z ...
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2answers
661 views

short exact sequence of holomorphic vector bundles splits but not holomorphically, only $C^{\infty}$

If there is a short exact sequence of holomorphic vector bundles, $$0 \overset{a_1}{\to} W \overset{a_2}{\to} V \overset{a_3}{\to} F \overset{a_4}{\to} 0,$$ then one can expect a $C^{\infty}$ ...