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

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Convert line parametrization into two equations

Consider the following parametrization on $\mathbb{R}^3$ $$g(t) = (t^2,t\cos(t),t\sin(t))$$ This is a line, and as such can be characterized by two equations. I already found the first one to be ...
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1answer
12 views

how to prove that $C^{k}$ map does not depend on choice of the charts

I was reading an article about Manifolds.They have defined a $C^{k} $ function in the following way : Let $M$ and $N$ are two $C^{k}$ manifolds of dimensions $m$ and $n$ respectively.A continuous ...
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0answers
16 views

$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
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0answers
19 views

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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0answers
18 views

Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if ...
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0answers
8 views

How to define this space? (matrix of coordinates)

We will let $F$ denote an arbitrary field such as the real numbers $R$ or the complex numbers $C$. For any positive integer $n$, the space of all $n$-tuples of elements of $F$ forms an ...
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1answer
22 views

Oriented atlas on a circle

I'm trying to find an oriented atlas on the circle $S^1$, i.e., I want to find an atlas for $S^1$ such that for any two overlapping charts $(U,s)$ and $(V,t)$ of the atlas, the derivative $d s/d ...
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1answer
21 views

Integration on k-1 form

If $\omega$ is a $k-1$ form on a closed $k$-dimensional manifold $M$ then $\int_M d \omega = 0$. I'm looking for a short proof to this problem, would Stokes be helpful?
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1answer
22 views

How transversality condition implies that a value is regular?

Currently I am self-learning some manifold theory and just come across concept of functions transverse to submanifolds. It seems that this concept is used a lot for proving regularity of values, but I ...
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1answer
23 views

Find a nontrivial bundle of $S^1$ with fibre isomorphic to $\mathbb{R}^n$

Show that such a nontrivial bundle exists for every $n\in\mathbb{N}$. I don't really have any useful ideas here. I'm not sure if there is a general approach I should be taking or if there is just a ...
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1answer
29 views

analysis on manifods

Let $M$ be a compact oriented $k+l+1$ dimensional manifold without boundary in $\mathbb R^n$. Let $\omega$ be a $k$-form and let $\eta$ be an $l$-form, both defined in an open set of $\mathbb R^n$ ...
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1answer
25 views

Why is $\int_{X} d(\alpha \wedge *\bar{\beta})$ zero on a compact hermitian manifold?

I am reading the book Complex Geometry - An Introduction by Huybrechts. In proving Lemma 3.2.3 that $\partial$ and $\partial^*$ are formal adjoints to each other, he mention that the following ...
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2answers
36 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
3
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0answers
16 views

Counterexample for the density of smooth functions in Sobolev spaces on a manifold

I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds $M,N$ the space $C^\infty (M,N)$ is dense in $L^p (M,N)$ if $\dim(M) > p$. (The ...
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0answers
23 views

What is concretely a vector field?

Let $M$ be a manifold and $TM=\coprod_{x\in M}T_xM$ be the tangent bundle. By definition, a vector field is an application \begin{align*} X: M&\longrightarrow TM\\ m&\longmapsto X_m\ni T_mM ...
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1answer
20 views

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for $k = n$ and $k = n-1$?

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for: (1) $k = n$, and (2) $k = n - 1$. Poincaré Duality tells us that for $M$ a closed ...
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1answer
17 views

Prove the boundary is a compact 1 manifold

A closed surface with boundary is a compact connected topological space $B$ with the property that each point $p \in B$ has an open neighborhood $U$ homeomorphic to either: $\{(x, y) \in ...
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3answers
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Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
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2answers
84 views

Examples of Manifolds such that $\chi (X)=-3$

I am trying to come up with an example of a closed oriented manifold with euler characteristic equal to $-3$. I have tried to use $\chi (\underbrace{T^2\mathbin{\#}\cdots \mathbin{\#} T^2}_{\text{$g$ ...
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1answer
26 views

Proof formalization help: Given a vector $u$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at 2 points.

Proof formalization help: Given a vector $u$ of Euclidean length $1$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at at least 2 points. I've thought about the ...
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0answers
42 views

Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
2
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2answers
45 views

Showing that the set of semi-orthogonal matrices is a $C^\infty$ submanifold

For $k, n \in \mathbb{N}$ with $k ≤ n$, we define $$S_{n, k} = \{X \in \mathbb{R}^{n \times k}: X^t X = I_k\}$$ where $I_k$ is the identity matrix of rank $k$. I want to prove that $S_{n, k}$ is a ...
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1answer
29 views

Under which additional hypothesis are open maps locally injective

Recollection of basic definitions: We recall the basic definitions that a continous map of topological spaces $f : X \to Y$ is open if $f(U)$ is an open subset of $Y$ whenever $U$ is an open subset ...
2
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1answer
51 views

Is the cuspidal cubic $\{y^2 = x^3\} \subset \Bbb R^2$ not smooth?

Cuspidal cubic $y^2=x^3$ in $\Bbb R^2$ "seems to be not smooth" intuitively because its pictured graph has a cusp at the origin. But I read from book that it is a smooth manifold. I feel so confused. ...
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12 views

explaination of the metric tensor on another manifold?

In skew -product decomposition the following features are observed :- 1.the Riemannian Manifold $(M,g)$ has a product form of $$M=R\times \Theta$$ Where $\Theta ,R $ are connected $C^\infty$ ...
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26 views

the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
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0answers
34 views

any open set in $\mathbb{R}^n$ is a $n$ dimensional manifold

I am trying to show this using the definition: M is a k-dimsensional submanifold of $\mathbb{R^n}$ if for all $x \in M$ the following condition holds: There exists an open set $U \subset ...
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$\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
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2answers
64 views

What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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0answers
20 views

What is the motivation for wanting to remove redundant terms that arise from the wedge product of two multilinear functions?

I am currently working through Loring Tu's "An Introduction to Manifolds," specifically the section in which the exterior algebra of multicovectors is introduced, and I am having trouble grasping the ...
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20 views

Why is $H_{DR}^p(M,\mathbb{C})\cong H_{DR}^p(M,\mathbb{R})\otimes_\mathbb{R}\mathbb{C}$

This question is related to my previous question. The answers to that question inspired a new question, namely For a complex manifold $M$, why is $H_{DR}^p(M,\mathbb{C})\cong ...
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1answer
42 views

Two proof of Petersen's 'Manifold'

Picture below is from the 5 page of Petersen's Manifold. First, why diffeomorphism is forced to be a lieanr isomorphism? The define of diffeomorphism accords to Wiki. Second , what space the point ...
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1answer
42 views

Poincaré Duality in Middle Dimension

I am reading a paper that states the following theorem without proof: Poincaré duality in middle dimension: Let $M$ be a connected oriented manifold of even dimension $2d$. Then the cup product ...
4
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1answer
35 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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1answer
33 views

Left-invariant vector fields on the circle $S^1$

I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...
1
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1answer
77 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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0answers
25 views

Showing that a set is a smooth submanifold

How do I show that the solution set to the equations $x^3+y^3+z^3=1$ and $z=xy$ is a smooth manifold? Regular value theorem doesn't apply here and I don't know how to construct an atlas for it.
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1answer
40 views

Derivative of map $f: S^n \to \mathbb{R}P^n$ is an isomorphism

I'm trying to show that the map $f: S^n \to \mathbb{R}P^n$ given by sending a unit vector $x$ in $S^n \subset \mathbb{R}^{n+1}$ to the line spanned by $x$ in $\mathbb{R}P^n$ has injective derivative. ...
0
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1answer
43 views

Why is a metric?

I have a question about tensors and metrics: Let $M=\{(t,x,y,z)\in \mathbb{R}^4: t>-1 \}$ and let $g=(1+t)dtdx+dy^2+dz^2$ Show that g is a metric on $M$. I did the next, I have the basis $\{ ...
0
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1answer
25 views

On computing the Differential of a Smooth Map

In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$ $$D_A F(B) = ...
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1answer
34 views

Why is $\phi^* g = g$ a PDE for a pseudo-Riemannian metric $g$ on a manifold?

Given a (locally trivial) bundle $\pi: E \to M$ a PDE of order $k$ is usually defined to be a submanifold of the jet-bundel $J^k(E)$. Now assume $E = M \times M$ and $\pi$ is the projection on the ...
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1answer
41 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
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0answers
69 views

Which homology groups of a closed orientable 6-manifold can be isomorphic to $\mathbb{Z}^3$?

List all $i$ for which there is a closed orientable $6$-manifold $M$ with $H_i(M) =\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ I am working on an old exam problem and this one stumped me. ...
4
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1answer
72 views

Why is the image of the implicit function in the implicit function theorem not open?

We have a continuously differentiable function $f$ from $\mathbb{R}^{n+m}$ to $\mathbb{R}^n$, and we find a continuously differentiable function $g$ which maps points from $\mathbb{R}^m$ into ...
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1answer
32 views

Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
2
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1answer
55 views

Question about connections on the dual bundle.

Let $E \to M$ be a vector bundle with connection $\nabla$. Extend $\nabla$ to $E^*$ and $E^* \otimes E$ in the regular fashion. Is $\text{Id} \in E^* \otimes E$ necessarily parallel?
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19 views

Quadrature over a (smooth, compact, convex, etc.) Riemannian manifold

Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ...
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32 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
2
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1answer
24 views

Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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1answer
47 views

Every $\mathcal{C}^1$ manifold can be made smooth?

I heard of a theorem saying that each $\mathcal{C}^k$-manifold with $k\geq 1$ can be made into a smooth manifold, i.e. $\mathcal{C}^{\infty}$ (by restriction of the atlas). However, I cannot find ...