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

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1answer
51 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 ...
3
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2answers
62 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 ...
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0answers
53 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 $K$ ...
0
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2answers
119 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, ...
1
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1answer
99 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 ...
1
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0answers
22 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 ...
2
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1answer
41 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 ...
2
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1answer
57 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 ...
2
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1answer
68 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 ...
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0answers
20 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 ...
2
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0answers
36 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 ...
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0answers
28 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
19 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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0answers
27 views

Jacobian on manifolds

I'm trying to make sense of integrals of the form $$\int_\Omega L[D\psi_1(x), D\psi_2(x)]\ \mathrm dx$$ Where $\Omega \subset \mathbb R^d$ is a $p$-dimensional compact manifold with boundary. ...
0
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1answer
29 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 ...
4
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3answers
171 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 ...
1
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0answers
23 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 $f$ ...
1
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1answer
66 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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0answers
20 views

3-manifold with boundary and corners

In the literature I often come across the sentence "3-manifold with boundary and corners" but I am not sure what does that mean? To be more specific: - What types of boundary can a generic 3-manifold ...
2
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2answers
126 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
33 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
54 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 ...
1
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1answer
146 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 ...
0
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0answers
24 views

Torus linking and natural geometric on the torus

I have the three-dimensional manifold $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ which we can see it as the configuration space of 3 points on a ...
0
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1answer
37 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 ...
1
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1answer
43 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 ...
1
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1answer
60 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 $U_{\alpha}$, ...
0
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0answers
41 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 ...
4
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0answers
65 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 ...
2
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0answers
42 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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0answers
29 views

Equal integrals, $2k$ - forms, real analytic subvariety, set of its regular points

Could you tell me where I can find a proof of the following fact? Suppose $V$ is a complex analytic subvariety of an open subset $U \subset \mathbb{C}^n$ of pure dimension $k$. Let $V_r$ be the set ...
1
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1answer
40 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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0answers
34 views

Finding boundary coordinate chart

I need to calculate the boundary coordinate chart of the manifold with boundary $$M=\{(x,y,z)\colon x^2+y^2+z^2=1, z\ge0\}$$ If I define $U=\{ (u,v)\colon u^2+v^2\lt1 \}$ and ...
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2answers
60 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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0answers
46 views

Germ induced by a submanifold

I'm currently reading this article: STOKES' THEOREM ON REAL ANALYTIC VARIETIES by LUTZ BUNGART In it the author writes: "In the following, we let $W$ be a closed real analytic subvariety of an open ...
12
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1answer
130 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 ...
0
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2answers
22 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 ...
1
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1answer
43 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
88 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 ...
1
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1answer
59 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, ...
0
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1answer
27 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!
0
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1answer
31 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 ...
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0answers
27 views

Doubt in an expression of the vector field

I know that $$\{\dfrac{\partial}{\partial x_i}:i=1(1)n\}$$ is a basis of the $n$ dimensional tangent space $T_p(M)$ [Vector space over $\mathbb R$] at the point $p$ on $M.$ Again I came to know ...
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1answer
31 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 ...
2
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1answer
45 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 ...
4
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2answers
80 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
63 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 ...
0
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0answers
24 views

Local submersion theorem and $O_{3}(\mathbb{R})$

I was attempting to follow the proof of the local submersion theorem given in Differential Topology by Guillemin & Pollack in the case that $X = O_{3}(\mathbb{R})$ and $f(A) = AA^{T}$. I worked ...
3
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0answers
38 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?
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0answers
58 views

Munkres' Analysis on Manifolds and Differential Geometry

Will Munkres' Analysis on Manifolds prepare me for a text like John Lee's Introduction to Topological Manifolds and his Introduction to Smooth Manifolds text? Would one be able to successfully tackle ...