For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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Adjoint representation for matrix groups (Gauge theory)

This is a question in regards to an identity in Gauge theory. Let $\omega$ be the connection one form on a principal bundle $\pi:P\rightarrow M$ and let $A_{\alpha}:=s_{\alpha}^*\omega$ be the gauge ...
5
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1answer
51 views

Sections of associated bundles

Let $\pi:P\rightarrow M$ be a Principal bundle and $\pi_V:P\times_G F\rightarrow M$ be its associated bundle via the representation $\rho:G\rightarrow GL(V)$. Fact: $\Gamma(P\times_G V)\simeq\{f:P\...
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1answer
47 views

Winding number of differential curve

Consider the one-form $\omega$ on $\textbf{R}^2$\ {(0,0)} defined by $\omega$ = $\frac{xdy-ydx}{x^2+y^2}$ Let K $\subset$$\textbf{R}^2$\ {(0,0)} denote the positive x-axis. Let $\gamma$ : $[a,b]$ ->...
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0answers
128 views

Gradient in local coordinates on a manifold with Riemannian metric

Let $M$ be a smooth manifold with a Riemannian metric g : $TM\otimes TM$ -> R If f is a smooth function from M to R, the gradient of f with respect to g is the vector field $\nabla f$ defined by $df$=...
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0answers
41 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
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2answers
68 views

Exterior differentiation of one form on a smooth manifold

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I'm fine with the right side of the equation, ...
2
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1answer
94 views

Guillemin & Pollack's proof on Whitney embedding theorem

I am confused with a little detail in Guillemin & Pollack's proof on Whitney embedding theorem. Please see page 54 in their book "Differential topology". In the second paragraph of page 54, they ...
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1answer
33 views

Quotient of a cylinder by the product of a rotation and translation

Let $D$ be the unit disk, and $$f : D \times \mathbb R \rightarrow D \times \mathbb R$$ be defined by $$f(z,t)=(e^{2i\pi \alpha}z, t+1)$$ where $\alpha$ is an irrational number. Now consider $X = (D ...
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1answer
151 views

Finding coordinate charts from a torus (square with sides identified) to $\mathbb R^2$.

I have a torus which is thought of as the unit square with sides identified in the usual manner. I know that for every point on the square I have to find a neighborhood and a diffeomorphism to open ...
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1answer
60 views

Passage in a proof of a lemma

Here is a lemma and a proof given to me in class. Lemma If $M$ is a smooth manifold, $K\subseteq M$ a compact subset, $A\subset M$ an open set containing $K$< then there exists a compact-support ...
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2answers
76 views

$\Delta_L(\text{im}\,\delta^*_g)\subset\text{im}\,\delta^*_g$ and $\Delta_L\big(\text{ker}\,\text{Bian}(g)\big)\subset\text{ker}\,\text{Bian}(g)$?

Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The Lichnerowicz Laplacian of $g$ is the map \begin{...
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0answers
68 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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1answer
142 views

Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, D_1,...
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1answer
143 views

Integration of forms on non-simply connected manifolds

What I know is that closed forms are not exact on non-simply connected manifolds, so for instance, if $E$ is a closed form, then $dE = 0$ but $\int_\gamma E \neq 0$, where $\gamma$ is a non-...
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0answers
30 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
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1answer
70 views

Are knot complements prime 3-manifolds?

Well, that is basically the question. Is the complement of a knot, i.e. an smoothly embedded copy of $S^1$ in $S^3$ a prime $3$-manifold? Here I mean by prime: A connected $3$-manifold $M$ is prime ...
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1answer
97 views

Extension of Sections of Restricted Vector Bundles

Edit: Changing Question: There are two questions related questions: extending a smooth vector field extending a vector field defined on a closed submanifold I'm trying to answer a question which is ...
3
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1answer
495 views

Are open sets in $R^n$ homeomorphic to $R^n$?

I am working on exercise 1.1 and I think the way to do this would be to show that open sets are homeomorphic to $R^n$ or open balls in $R^n$. Is this even true? I'm not sure how to go about proving ...
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1answer
50 views

Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then $H_p:=\...
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1answer
266 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle $D:...
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1answer
58 views

What are all angle preserving linear operators on $\mathbb R^n$?

I´m working on Spivak's Calculus on Manifolds and I met this exercise. My immediate answer was 'all the rotations' but I can't explain why. Am I right? Can you give a hint or something to be able to ...
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1answer
37 views

Manifold that is Hausdorff and second countable

Why are we usually assume that a manifold $M$ has to be a Hausdorff space and Second countable ? Is it really hard to study smooth manifolds without making these assumptions?
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1answer
80 views

Definition of exterior derivative from a connection

I fail to see what is the meaning of the symbol $d_{\nabla}$ in (1.2) of http://arxiv.org/pdf/hep-th/9712042v2.pdf I know the meaning of that symbol in the context of forms taking values on some ...
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3answers
196 views

Are there such things as 'locally homogenous spaces'?

A Euclidean space has the property that every point has a neighbourhood that is homeomorphic to some neighbourhood of any other point. I'm not sure what the name of this property is - I thought it ...
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0answers
34 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
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1answer
154 views

Transverse submanifolds in product manifolds.

Suppose we have smooth manifolds $M,M',N$, a smooth map $f\colon M\rightarrow M'$ and a smooth submanifold $S'\subseteq M'\times N$, such that the projection $\pi_{M'}\colon S'\rightarrow M'$ is a ...
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2answers
152 views

Integrating over the two form

Let $A=(0,1)^2$. Let $\alpha:A\to\Bbb R^3$ be given by the equation $$\alpha(u,v)=(u,v,u^2+v^2+1)$$ Let $Y$ be the image set of $\alpha$. Evaluate the integral over $Y_\alpha$ of the 2-form $...
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1answer
52 views

Is the tangent bundle of a covered manifold a quotient manifold?

Given a covering manifold $\rho :\widetilde M \to M$ we know that $M$ can be thought of as the quotient space of $\widetilde M$ like so $M = \widetilde M /\ G$ where $G$ is the monodromy group (or ...
2
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1answer
100 views

How to “lift” a path to the tangent bundle?

Given a path $c: (-\epsilon,\epsilon)=I \to M$ in a manifold. Define $\widetilde c:I \to TM$ (a kind of "lift") as the curve $t_0 \mapsto [c(t+t_0)] \in T_{c(t_0)}M$. Is there a nice categorical ...
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1answer
214 views

Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19): Wirtinger's Inequality. Let $L$ be a complex linear space and let $M$ be a real even-dimensional ...
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1answer
60 views

Manifold learning: How should this method be interpreted?

I am trying to learn about manifold learning techniques; a family of dimensionality reduction methods in machine learning. According to this idea, there is a low ($d$) dimensional, hidden space where ...
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1answer
192 views

Locally Euclidean Hausdorff topological space is topological manifold iff $\sigma$-compact.

I'd like somebody to specify flaws in my outline of the proof of the above statement. I'm following the definition of topological manifold used in Lee's Introduction to Smooth Manifolds. (it is 2nd ...
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1answer
84 views

$H^{n}(M)$ where $M$ is compact, orientable and connected manifold

I need to show that if $M$ is compact, orientable and connected manifold of dimension $n$, then $H^{n}(M) = \mathbb{R}$. I saw that a possible proof is to take an atlas for $M$, with $U_{\alpha}$, $\...
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0answers
79 views

show that if $M \times \mathbb{R}^{n}$ is orientable than so is $M$

I need to show if $M \times \mathbb{R}^{n}$ is orientable than so is $M$, where $M$ is connected manifold. $R^{n}$ has standard orientation (determined by standard basis ) and by the assumption $M \...
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0answers
142 views

Can a differential k-form be integrated on a manifold that is not k-dimensional?

For example, can you integrate a 2-form on some curve, a 1-dimensional manifold, or some 3-dimensional manifold? I know that Stokes's Theorem states that if you integrate $\omega \in \mathcal A^{k-1}(...
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0answers
55 views

Help! How to derive the result related to Darboux derivative?

First, define Darboux derivative. There is one Lie group $G$ and one manifold $M$. Let $\phi:M\rightarrow G$ be a smooth map. The Darboux derivative $\Delta(\phi):TM \rightarrow M\times \mathfrak g$ ...
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1answer
93 views

How to show the one point compactification of a cylinder isn't a manifold.

I'm trying to show that the one-point compactification of a cylinder $C^*$ = C$\cup${$\infty$} isn't a manifold. The way I'm trying to show this is if $C^*$ is a manifold then if I take a ...
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2answers
80 views

Proving that $\Delta(M \times M)$ is a submanifold of $M \times M$

I am struggling to prove that $\Delta(M \times M) = \{(x,x) : x \in M\}$ is a submanifold of $M \times M$. A manifold M is a submanifold of N if there is an inclusion map $i:M \rightarrow N$ ...
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2answers
371 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
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2answers
26 views

Any neighbourhood of a point $x$ in a manifold $X$ ($\dim X \geq 2$) has a subneighbourhood $V$ of $x$ such that $V \setminus \{x\}$ is connected

What I want to show for this is that $V \setminus \{x\}$ is homeomorphic to some punctured ball, $B_m \setminus \{p\}$ (where $B_m$, $p \in \Bbb R^m$). And then since $B_m \setminus \{p\}$ is path-...
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1answer
64 views

Curves and tangent vectors in a manifold setting

Consider the following definition: ($M$ denotes a manifold structure, $U$ are subsets of the manifold and $\phi$ the transition functions) Def: A smooth curve in $M$ is a map $\gamma: I \rightarrow ...
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1answer
153 views

Manifolds, coordinate systems, books

Which books, say Lee's Introduction to Smooth Manifolds or Munkres' Analysis on Manifolds explains how the theory of a differentiable manifolds can be used to solve a problem that is expressed in a ...
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1answer
194 views

Extension Lemma for Smooth maps (Lee vs. Lee)

I've been reading Jeffrey Lee's, Manifolds and Differential Geometry and John Lee's, Introduction to smooth manifolds. In the first book (here, in page 31), after introducing partition of unity, ...
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1answer
45 views

Checking alternating tensors

How do I check that $$f(x,y)=x_1y_2-x_2y_1+x_1y_1$$ is an alternating tensor? I did check that f is a tensor, but how do I know if it is alternating by direct computation? Thanks in advance!
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1answer
78 views

Examples of mod 2 degree of f.

I'm reading Milnor's Topology from the Differentiable Viewpoint and I've just finished the chapter about degree mod 2. Since every two smooth homotopic maps $f,g$ must have the same degree mod 2 it'...
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1answer
46 views

Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued $k$-...
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1answer
96 views

Tangent space of tangent vector

Let $M$ be a smooth manifold. There's a (split) short exact sequence $$0\to T_aM\to T_v(TM)\stackrel {D_v}{\to} T_aM \to 0,$$ where $v\in TM$ and $a=\pi(v)\in M$. I'm trying to understand what this ...
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2answers
152 views

Principal bundles, connection forms and fundamental vector fields

Suppose $\pi:P\rightarrow M$ is a principal bundle, $\omega\in \Omega^1(P;\mathfrak{g})$ is the connection one form and $\sigma(\cdot)$ is the fundamental vector field associated to some vector field ...
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1answer
141 views

A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds. I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff. It seems easy to see that the ...
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0answers
46 views

A kind of uniqueness for the double of a manifold

Let $M$ and $N$ be two manifolds with the same boundary. If their doubles $D(M)$ and $D(N)$ are diffeomorphic, are $M$ and $N$ diffeomorphic?