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

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2
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1answer
22 views

Maps between Manifolds and Maximal Rank

I'm trying to prove a theorem from Olver's Applications of Lie Groups to Differential Equations. It's supposed to be an "easy" consequence of the Implicit Function Theorem but I honestly can't see ...
2
votes
1answer
54 views

Equivalent topologies on Real projective space $RP^{n}$

This is homework,so no answers please. Prove that the topology on $RP^{n}$ given by the standard smooth structure (lines through the origin in $\mathbb{R}^{n+1}/\{0\}$ and $\tau_{1}$) is equal to ...
2
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1answer
29 views

Tangent space to $\mathbb{R}P^{n}$

I could not find any other question here related to this. If I have missed out, then this could be voted as a duplicate(Sorry if it is!). I was just trying to figure out the tangent space to the ...
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1answer
42 views
+50

Codimension 1 homology represented by Embedded Submanifold

I'm looking for a reference for the following statement: Given an oriented manifold $M$ and a class $\xi \in H_{n-1}(M)$, there is some embedded oriented submanifold $F^+ \hookrightarrow M$ such that ...
1
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0answers
23 views

Given points $x,y$, and tangent vectors $v,w$, is there a diffeomorphism $x\mapsto y$ and $v\mapsto w$?

Suppose $M$ is a smooth manifold, and you're given two points $x,y$, and associated tangent vectors $v$ and $w$. I'm curious, does there exist a diffeomorphism $F$ such that $F(x)=y$ and $dF_x(v)=w$? ...
0
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1answer
27 views

Real/Complex Manifolds - Transition Maps

I'm trying to understand how real/complex structure is imposed on a manifold, especially the likes of smooth manifolds. I can read the definitions and work with them, but I want to understand ...
0
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2answers
36 views

connected sum of two surfaces

I was reading Massey's textbook on Algebraic topology and the author claims that if $S_2$ is a 2-sphere then $S_1 \# S_2$ is homeomorphic to $S_1$. I don't know why that is true and since I'm very ...
2
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1answer
40 views

Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
4
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4answers
752 views

Why surface of the sphere is not a Euclidean space?

In wiki, I see a description in manifold: "Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in ...
0
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1answer
37 views

Braid Groups on Manifolds

I am studying braid groups on manifolds and am getting confused. In a geometric definition, one needs to first choose a simple curve $\theta$ on a given manifold $M$ and well-ordered points ...
2
votes
2answers
288 views

Is a topological space with a Minkowsi metric a topological manifold?

The definition of a topological manifold from Wikipedia: tm defines it as a topological space which locally looks like Euclidean space. But what about a topological space that uses the Minkowski ...
1
vote
1answer
26 views

Knots as boundaries

Boundary of a 2-manifold is a closed curve (or a set of closed curves), so I was thinking of reversing this process. In 2D space, a simple closed curve in a plane can be thought of as a boundary of ...
0
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3answers
54 views

What exactly is a 0-form?

From what I understand, a k-form in the real numbers is essentially a mapping $\mathbb{R^k} \rightarrow \mathbb{R}$, but I can't seem to find a corresponding definition for a "0-form". Wikipedia seems ...
1
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1answer
34 views

Simultaneous coordinate representation of a submanifold and its sub-submanifold

Suppose $Z\subset X\subset Y$ are manifolds and $z \in Z$. Prove that there exist an independent function $g_1,...g_l$ on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W =\{y\in ...
0
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0answers
17 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
10
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2answers
178 views

Ways to think about vector bundle

I'm studying manifold theory and I've got to the point of discussing the definition of a vector bundle. The definition is quite long and a bit confusing and I was wondering if someone with a bit more ...
4
votes
3answers
303 views

What is the codimension of matrices of rank $r$ as a manifold?

I'm reading through G&P's Differential Topology book, but I hit a wall at the end of section 4. There is a result stating The set $X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$ is a ...
0
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1answer
21 views

Geometric translation of a theorem about stability of equilibrium point

In the book Nonlinear Systems by Hassan Khalil, there is a theorem about the stability of equilibrium point ‎‎ which asserts that : Theorem :‎ Let‎ ‎$X = 0 $ ‎be an equilibrium point for‎ ‎$‎\dot{x} ...
6
votes
1answer
163 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
0
votes
1answer
35 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
2
votes
1answer
35 views

trace map is continuous

Prove that $tr: M_n(k)\to k$ is continuous. I did continuity of determinant map using induction, but how to prove trace map is continuous. please give a thorough answer. My analysis is not too good. ...
0
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0answers
17 views

Orientation of the intersection of manifolds

From Guillemin and Pollack Differential Topology: Compute the orientation of $\mathbf{X}\cap\mathbf{Z}$ in the following examples by exhibiting positively oriented bases at every point: a) ...
1
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0answers
31 views

Compact manifold with smooth structure

Is the following surface smooth and compact, when all its partial derivatives are continuous? How to tell about self-intersections without visualization? $x=\cos(a) + \cos(a + b) + \cos(a + b + ...
1
vote
2answers
365 views

Symmetric, upper triangular, diagonal and null-trace matrix spaces: are they manifolds?

I have to prove that to each of following classes of matrices can be given a manifold structure: symmetric (denoted with $\mathcal{S}$) upper triangular diagonal null trace. I am interested in ...
0
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0answers
28 views

clear statement about the relation between curvature and rotating vectors along a loop

in my research, I need to understand the relation between the formal definition of Riemann curvature tensor ($R_{jkl}^i$)and the ``vector rotation" approach: parallel transport a vector along a loop, ...
1
vote
1answer
41 views

Reference request: Measure theory and/or manifolds [duplicate]

I have never encountered measure theory or manifolds yet, despite being close to my third year university level. Any texts for either or both of these subjects would be greatly appreciated.
-1
votes
1answer
21 views

Identity concerning push forward of two vector fields

How would you prove the identity $\displaystyle \frac{\partial}{\partial s}\Psi_{s^*} \mathbb{X} = (-1)L_{\mathbb{Y}}\Psi_{s^*}\mathbb{X}$ where $\Psi_{s}$ is the flow of $\mathbb{Y}$ and ...
1
vote
1answer
38 views

tangent bundle and normal bundle

I have a problem about tangent bundle. It is known that the tangent bundle of most manifolds is not trivial: for example, the tangent bundle for $S^2$ is not $S^2\times \mathbb{R}^2$. However, for a ...
8
votes
1answer
108 views

Why are Banach manifolds not so popular?

Why are Banach and Frechet manifolds studied not even remotely as much as Euclidean manifolds? I assume like many other mathematical subjects, theory of manifolds has been developed much more than the ...
0
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0answers
14 views

Show that $M$ is not equivalent to $O_k$.

Assume that a model hypersurfaces is described by $$O_k=\{(z,w)\in \Bbb C^2 \mid v=|z|^k\}\tag 1$$ and a real analytic hypersurface: $$M=\left \{(z,w)\in \Bbb C^2 \mid ...
4
votes
1answer
36 views

Converse to the Jordan-Brouwer separation theorem

By the Jordan curve theorem, if $C \subset S^2$ is (the image of) a simple closed curve, then $S^2 \setminus C$ has precisely two connected components. This statement admits the following "converse". ...
0
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0answers
18 views

Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
4
votes
2answers
138 views

Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$

I noticed that there exists a (in some sense) better definition of the tangent space via the dual of a certain quotient algebra which is easier to work with in some cases. This however only works for ...
9
votes
0answers
100 views

$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
0
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1answer
14 views

Writing nonautonomous systems as autonomous systems

Apparently any mth order nonautonomous system is equivalent to a first order autonomous system in higher dimensional space. How does this work in practice? I would you write $\displaystyle ...
0
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1answer
123 views

Liouville form on the cotangent bundle

a) Let $(U,\phi) = (U,x^1,\dots,x^n)$ be a chart on a manifold $M$, and let $(\pi^{-1}U,\tilde {\phi}) =(\pi^{-1}U,\tilde {x^1},\dots,\tilde {x^n},c_1,\dots,c_n)$ be the induced chart on the ...
0
votes
1answer
30 views

Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
0
votes
3answers
52 views

A proof of compactness, connectedness of real projective space

I need a reference for a complete proof of the below theorem: Let $RP^n$ be $n$-dimensional real projective space. Then $RP^n$ is a compact, connected manifold. (Consider the standard topology over ...
0
votes
1answer
15 views

Equivalence relation of differential forms

My notes claim that $\displaystyle d\omega (x) = \frac{1}{k!} d\omega_{i_1\cdots i_k} \wedge f^{(i_1)}\wedge\cdots\wedge f^{(i_k)}$ is equivalent to $\displaystyle d\omega(x) = \frac{1}{k!} ...
0
votes
1answer
38 views

Help explain linear algebra/differential calculus theorem in simpler terms.

On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I ...
0
votes
1answer
40 views

Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton $$A\sim ...
0
votes
2answers
44 views

Differential identity and wedge products

Apparently $dx^{i_1} \wedge ... \wedge dx^{i_k}=d(x^{i_1}dx^{i_2}\wedge ... \wedge dx^{i_k})$ which I cannot see proved anywhere in my notes. It just stated as if it is obvious which I don't believe ...
2
votes
1answer
71 views

Riemannian metric and geodesic

For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$. I am not sure about the ...
2
votes
2answers
67 views

Part of proof that $d^2\omega=0$

The following comes from the proof in differentiable manifolds that $d^2\omega=0$. Let $f$ belong to the set of $0$-forms. From definition I have that $\displaystyle df = \frac{\partial f}{\partial ...
2
votes
1answer
95 views

Bott connection

Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication ...
3
votes
3answers
60 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
4
votes
1answer
54 views

Existence of critical points of $f:\mathbb{C} -\{0,1\}\to \mathbb{R}$

I am trying to show that an smooth, proper map, $f:\mathbb{C} -\{0,1\}\to \mathbb{R}$ has a critical point. My attempt was to suppose there are no critical points, then the preimage of every point is ...
6
votes
1answer
102 views

Is T($S^2 \times S^1$) trivial?

How would I find out if T($S^2 \times S^1$) is trivial or not? Using the hairy ball theorem I can show that T($S^2$) is not trivial, and it is straight forward to show that T($S^1$) is trivial. ...
4
votes
1answer
101 views

Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
1
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2answers
32 views

Definition of a coordinate vector bundle

Consider the following definition of a coordinate vector bundle. Let $M$ be a smooth manifold of dimension $m$, and $\{(f, U_f)\}$ an atlas of compatible charts for $M$. A smooth coordinate ...