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

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If $\omega$ is a differential $4$-form on a $10$-manifold $M$ then $\omega \wedge d\omega$ is exact

Let $\omega$ be a differential $4$-form on a $10$-manifold $M$. I am trying to show that $\omega \wedge d\omega $, which is a $9$-form, is exact. Clearly $\omega \wedge d\omega$ is closed, because $...
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7 views

calculating the colored HOMFLY polynomial for rational links

There are simple recursive ways to calculate the Jones, Alexander and HOMFLY polynomial for rational links. Is there such a formula, or any known way at all, to calculate the the colored HOMFLY ...
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27 views

How to obtain real vector from abstract tangent vector in the case of the manifold $\mathbb R^n$

I know that for every $p\in\mathbb{R}^n$ the map \begin{align} \Phi_p\colon\mathbb{R}^n&\to T_p\mathbb{R}^n\\ v&\mapsto D_{v,p} \end{align} is an isomorphism, where \begin{align} D_{v,p}\...
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Torus bundle over sphere

Let $G$ be a Lie group and let $T\cong\mathbb C/\mathbb Z^2$ be a compact complex torus. What is an example of a nontrivial $G$-fiber bundle $$T\hookrightarrow E\to S^2\; ?$$
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Structure preserving maps between manifolds with boundary

While learning the basics about smooth manifolds with boundary in this semesters' course about analysis on manifolds, there's a seemingly basic property I didn't find anywhere. Namely I want to ...
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1answer
28 views

Confusion about the Definition of Smooth Functions on a Manifold

I am slightly confused about the definition of smooth functions on a smooth manifold given in An Introduction to Manifolds by Loring Tu (Second Edition, page no. 59). The definition is given below. I ...
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1answer
33 views

Definition of topological manifolds with dimension zero/locally euclidean of dimension zero

A topological $n$-manifold $M$ is locally Euclidean of dimension $n$ (each point of $M$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^n)$. But what does locally Euclidean ...
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1answer
46 views

$S^2$ as a totally real submanifold of $\mathbb{CP}^1\times \mathbb{CP}^1$

Can the sphere $S^2$ be embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a totally real submanifold?
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The manifold hypothesis and classification of datasets

In the picture below, figure 1 and 3 show a 2-dimensional manifolds while figure 2 and 4 show those same manifolds embedded in a 3-dimensional space. My question is: In the pair of figures 1 and 2, is ...
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1answer
25 views

Average of a tensor with respect to a group action

Consider a smooth manifold $M$ (assume it boundary-free and orientable) and a tensor field $\mathcal{G}\in\Gamma(\otimes^hTM\otimes^kT^*M)$. Let $\Phi:\mathbb{T}^p\times M \rightarrow M$ be a torus ...
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1answer
47 views

Finding stable and unstable manifolds

Consider the system $x ^ { \prime } = 4 x + 2y ^ { 3 } \\ y ^ { \prime } = - 3 x$ Question: Find the stable and unstable manifolds around the fixed point $(0,0)$ and and sketch the phase portrait ...
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30 views

Thurston Compactification

Sorry in advance for my English. I'm studying the Thurston compactification from the Jean-Pierre Otal's book "The Hyperbolization Theorem for Fibered 3-Manifolds". I have a question, what $\mathbb{...
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23 views

Tu's An Introduction to Manifolds - Section 26.2 Cohomology of a circle, tabular form.

I'm trying to understand how to use the Mayer-Vietoris sequence to compute Cohomologies. There's a small chapter in Tu's Introduction to Manifolds explaining the basics, with some basic theory. More ...
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42 views

Topological Manifold which does not admit Riemann surface structure?

I have a question motivated by some examples in section 4.2 of Schlag's A Course in Complex Analysis and Riemann Surfaces. Example 1. Any smooth, orientable two-dimensional submanifold of $\mathbb{R}...
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32 views

Hairy ball theorem for $S^2$ [duplicate]

It is well-known that there is "no" nowhere vanishing continuous tangent vector field on $S^2$, by the so-called Hairy-ball theorem. But then, is there a continuous tangent vector field on $S^2$ which ...
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22 views

Coordinate curve depends on $t$

I have a basic problem to understand the coordinates $(x)$ on a manifold $M$. The coordinate vector is $$\frac{\partial}{\partial x^j}$$ which says that the tangent vector depends on $x^j$ but the ...
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1answer
24 views

Coordinate vector and curve

How can I see that the $j$-th coordinate vector $$\frac{\partial}{\partial x^j}$$ on a manifold is the velocity vector to the $j$-th coordinate curve parametrized by $x^j$. Both, by velocity and ...
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19 views

Book definition verification of boundary manifolds

Let $M$ be an n-dim topological manifold with boundary, A chart for M is a pair $(U,\phi)$ where $\phi: U\rightarrow \mathbb{R}^n$ is a map such that $\phi$ is a homeomorphism onto an open subset of $\...
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Boundary points and manifolds

Let $M$ be an $n$ manifold with boundary. Fix $p\in U$. Let $(U,f)$ be a boundary chart, for which $f(p)\in f(U)\cap \partial{\mathbb{H}^n}$. Then, for any other boundary chart $(V,g)$, such that $p\...
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16 views

Parametrizing a manifold

I have a conceptual question about parametrizing a 2-manifold in $\mathbf R^3$ and computing its area/volume. If there is anything that I didn't phrase correctly, I would be glad to change it. What ...
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15 views

The geometric of a.e. differentiable function [closed]

For smooth functions, we can explore their geometrics by a differential manifold or Riemann manifold. I wonder are there math tools for the geometrics of broader cases, such as a.e differentiable ...
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34 views

“Linear approximation” to topological sheaf $\mathrm{Imm}(-,N)$

I'm reading these notes Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$ If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element ...
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19 views

Every closed orientable surface is Riemann surface

I want to prove that every closed orientable surface is a Riemann surface i.e. every closed orientable surface admits a complex structure. Several proofs are available which make use of classification ...
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Hint in an exercise, manifold and its dimension

I'm triynig to solve this exercicse but I have a trouble: I don't know what is the dimension of $G$, it is finite or not? but I suppose the base is $(x_1,...x_n)$ so I can wrote any vector as follows ...
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1answer
30 views

Set of orthogonal matrices is a compact manifold

The following is a problem from Munkres's Analysis in Manifolds. Problem: Let $\mathcal O(3)$ denote the set of all orthogonal 3 by 3 matrices, considered as a subspace of $\mathbf R^9$. (a) ...
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1answer
51 views

Simplicial Homology of Matrix Groups

It is a well known fact that the matrix groups $GL_n(\Bbb R), SL_n(\Bbb R), \dots$ can be considered as submanifolds of $\Bbb R^{n^2}$. I did not yet attend a lecture on Lie groups, so I donā€™t know ...
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Solid torus is a 3-manifold.

I'm working on a problem from Munkres' Analysis on Manifolds, where I must show that a solid torus is a 3-manifold, and that the boundary of this manifold is the torus $T$. Letting $g$ be the ...
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May's proof - An $R$-orientation of $M$ determines an $R$-orientation of $\partial M$

In section 21.4 of May's A Concise Course in Algebraic Topology, there is a proposition that an $R$-orientation of $M$ determines an $R$-orientation. The proof is not so short, so I leave the link of ...
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1answer
20 views

lie bracket is vanish iff translation is commute

I want to prove following claim: Given two vector fields $X$ and $Y$ on smooth manifold $M$, $[X,Y]=0$ if and only if $\Phi^X_t \circ \Phi^Y_s \circ \Phi^X_{-t} \circ \Phi^Y_{-s}(q)=q$ for $\forall ...
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54 views

Rational cohomology of punctured closed non-orientable manifold.

Let $M-\{p\}$ be a punctured closed non-orientable even-dimensional manifold. If $M=\mathbb{R}{P}^{2}$ then $\mathbb{R}P^{2}-\{p\}$ is an open Mobius strip. This implies that $H^{*}(\mathbb{R}P^{2}-\{...
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The function $R(x)=rank(Df(x))$ is locally constant on $\Omega$, i.e. it is constant in a neighbourhood of every point $x \in \Omega$.

Let $f:\Bbb R^n \to \Bbb R^n$ be a mapping of class $C^1$. Prove that there is an open and dense set $\Omega \subseteq \Bbb R^n$ such that the function $R(x)=rank(Df(x))$ is locally constant on $\...
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Reason for defining $df(v)=D_vf$

I'm reading "A Visual Introduction to Differential Forms and Calculus on Manifolds", specifically the section on equivalence of directional derivatives to vectors acting as operators on functions ...
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1answer
17 views

Showing that the differential is an immersion

If $f: X \rightarrow Y$ is an immersion of smooth manifolds, then show that $df: TX \rightarrow TY$ is also an immersion. The definition of immersion(when dim$X <$ dim$Y$) that I have is that for ...
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1answer
50 views

Proving $R_{ij;m}=g^{kl} R_{ikjl;m}$.

In the coordinate $\{x^i\}$, the Riemann curvature tensor can be written as $$ R=R_{ijkl}\,dx^i\otimes dx^j\otimes dx^k \otimes dx^l $$ and the Ricci curvature can be written as $$\text{Ric}=R_{...
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Show that S is an embedded $k$-submanifold of $M$

Assume $M$ is a smooth manifold and $S$ is a subset of $M$ such that each point $p\in S$ has a neighborhood $U \subseteq M$ such that $S\cap U$ is an embedded $k$-submanifold of $U$. I want to show ...
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33 views

Exact one form on non-simply-connected manifold

Let $M$ be a smooth $n-$dimensional manifold and $\alpha\in\Lambda^1(M)$ a smooth $1-$form. Let $N\subset M$ be an embedded submanifold where $\alpha|_N=0$. If $d\alpha=0$ on the whole $M$, can I say ...
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59 views

Question about James Munkres's “Analysis on Manifolds”, p.112, Theorem 14.1

I'm an engineer who self-studies pure math in my free time, so I apologise in advance if this is a silly question. I am currently reading "Analysis on Manifolds" by James Munkres. Theorem 14.1 states ...
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1answer
32 views

Subjects needed to learn Manifolds

The way I go about learning mathematics by my self is the following: I set up a goal, for example, the most recent one was "Complex analysis", and then I learn my way up to my goal, for example, what ...
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1answer
30 views

Riemannian Exponential Map is a Homeomorphism outside the Cut Locus

Let $M$ be a connected and complete Riemannian manifold. The Hopf-Rinow Theorem guarantees that $Exp_p$, for any $p \in M$, is defined on all of $T_p(M)$. Further, this map is a diffeomorphism on a ...
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Does Alexander duality hold for compact, orientable Homology sphere?

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...
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1answer
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Regarding equivalent definitions of Euclidean Submanifolds in Gallot, Hulin and Lafontaine's book Riemannian Geometry

In the book Riemannian Geometry by Gallot, Hulin and Lafontaine, a proposition which characterises equivalent definitions of submanifolds is given as follows: 1.3 Proposition The following are ...
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37 views

Notation in differential forms

In a paper I've found a repeated notation with differential forms and, since I have never seen it before, my question is if there is a misprint or if it is an operation I don't know. I have a smooth ...
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19 views

Levi-Civita connection from idempotents

Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
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1answer
69 views

Argument in a proof for scalar maximum principle

I'm trying to understand how an assertion made in the proof of the scalar maximum principle follows from the compactness of the manifold we're working with. The situation is as follows: I ...
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1answer
26 views

On the geometric realization of a finite abstract simplicial complex which is connected, orientable $3$-manifold without boundary

Let $\Delta$ be an abstract simplicial complex on finitely many vertices and $|\Delta|$ be it's geometric realization. (https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex) If $|\Delta|$ is ...
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1answer
70 views

Second derivative test (and sign of laplacian at critical points) for manifolds

I'm trying to understand in more detail some of the justifications for a proof of the second derivative test for Riemannian manifolds, given below: I've never seen the Laplacian interpreted as an ...
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19 views

Prove that the boundary orientation of $S^k = \partial B^{k+1}$ is the same as its preimage orientation

I would like to verify if my approach to this problem is the correct one or not. This problem is from "Differential topology" by Victor Guillemin and Allan Pollack . More specifically is the problem 3....
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1answer
32 views

Find the rate at which distances decrease in stereographic projection

I want to map a 3D space onto the inside's surface of a sphere. The 3D space is represented by points (x,y,z) where the z axis is the height. The first thing I did was to use the following equation ...
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1answer
57 views

Equivalent definitions of the Möbius band

In Manfredo do Carmo's Riemannian Geometry, the Mƶbius band is defined as the quotient of the cyllinder $S = \{ (x, y, z) \in \Bbb{R}^3 \ : \ x^2 + y^2 = 1, \ - 1 < z < 1\}$ by the group $\{A, ...
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existence of a transverse homotopic function - example

I would like to find an example for the following theorem: Let $f: M \to N$ be a smooth map. There exists a map arbitrarily close to $f$, and homotopic to it, which is transverse to $Z$. In the ...

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