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

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4
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1answer
49 views

Does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point?

As the question suggests, does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance. To be clear, I'm using the statement of Brouwer's Fixed-Point ...
5
votes
0answers
55 views

Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
4
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2answers
50 views

Intersection of topological manifolds.

A condition for the intersection of two smooth manifolds to be a smooth manifold is that they intersect transversally. Is this only an obstruction because of the smooth structure? Question: Is the ...
2
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2answers
43 views

Compute the tangent space at the unit matrix

Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$ My attempt: I think I have computed the tangent space in the case that $M=SL(n)$. ...
0
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2answers
41 views

Is Fermat's theorem about local extrema true for smooth manifolds?

Let $M$ be a smooth manifold and $f\colon M \rightarrow \mathbb{R}$ a smooth function. If $p\in M$ is a local extremum of $f$, does $p$ have to be a critical point?
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1answer
20 views

Equivalence class of differential structures.

Given $X$ a manifold, two smooth atlas are equivalent if the union of them forms an smooth atlas. I am trying to prove that this is a equivalence relation, but I am having troubles proving that ...
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0answers
34 views

Looking for a reference that explains connections and curvature by double tangent space

I'm looking for a book or a set of lecture notes on differential manifolds that explain connections (Levi-Cevita connection, prinicipal connections) and curvature on an abstract manifold from the ...
2
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0answers
31 views

Show that a subset is a submanifold of $ \mathbb{R}^4$

Given the functions $f_i : \mathbb{R}^4 \to \mathbb{R}, i = 1, 2, 3$ with $f_1(x) = x_1x_3 - x_2^2$ $f_2(x) = x_2x_4 - x_3^2$ $f_3(x) = x_1x_4 - x_2x_3$ show that $M:= \{ x \in ...
4
votes
2answers
90 views

Gluing diagrams: is it possible to glue a surface with itself in the same point? how is the diagram drawn?

I am learning the basic concepts of Topology, and playing now with the gluing diagrams (describing the fundamental domain of a topological space), this is an excerpt of a basic description I took from ...
1
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0answers
42 views

Differential forms defined by integration

Let $\omega_1,\omega_2$ be two n-forms on a $n$-dimensional manifold $M$. Now, imagine we have for every open $N \subset M$ that $$\int_{N}\omega_1 = \int_N \omega_2.$$ Can anybody show me how to ...
5
votes
1answer
54 views

Why can't that be an uncountable union?

I'm reading part of Lee's Introduction to manifolds. I have come to the following proposition. The proof then continues, and I will read the rest shortly. I was just wondering: why can't ...
5
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0answers
46 views

Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and ...
0
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0answers
35 views

Reference for theorems in Hirsch

In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ ...
3
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1answer
70 views

I am confused by the different definitions of manifolds.

I'm currently learning manifold from Do Carmo's Riemannian Geometry. This is his definition of differentiable manifold: But this is different from what I saw in wiki: A differentiable manifold ...
2
votes
1answer
95 views

What exactly is a tangent vector?

I'm a physics student with a very ghetto understanding of the mathematics I use. I'm trying to learn a little more about very basic topology, manifolds, and Riemannian geometry. I'm using Nakahara's ...
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0answers
31 views

Ramanan's definition of differentiable function

In his book Global Calculus, Ramanan defines a differential manifold as follows: What is meant by condition (b)? Is $\mathcal A$ simply a subsheaf of the sheaf of real valued continuous functions ...
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0answers
27 views

Zeroes of differential form embedded

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ ...
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0answers
16 views

Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
2
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0answers
45 views

Chern-Gauss-Bonnet theorem for even-dimensional manifolds with boundary

On the wikipedia page for the Chern-Gauss-Bonnet theorem it states that there is a generalization of the theorem for even-dimensional manifolds with boundary, but does not provide the relevant theorem ...
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0answers
24 views

Leray-Hirsch theorem for cohomology modulo torsion

Suppose $X,Y$ are smooth manifolds, $H^*(X,\mathbb{Z})$ is finitely generated. (*)Why do we have isomorphism modulo torsion: $H^n(X\times Y,\mathbb{Z})=\oplus_{p+q=n}H^p(X,\mathbb{Z})\otimes ...
3
votes
1answer
55 views

Is every topological manifold completely metrizable?

Is every (second-countable) topological manifold completely metrizable? It is known that every smooth manifold possess a complete Riemannian metric, hence in particular it is completely metrizable, ...
1
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1answer
51 views

Multivariable Calculus: Manifolds

Problem Let $M$ be the set of all points $(x,y) \in \mathbb{R}^2$ satisfying the equation $$xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 $$ Prove that $M$ is a manifold. What is the dimension of $M$? ...
2
votes
2answers
32 views

Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
5
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0answers
53 views

Higher-Order Differential Operators as Vector Fields

On a $C^\infty$ manifold $M$, one can produce the tangent space $T_p M$ at a point by equivalence classes of tangents to smooth curves through the point $p$. When realised this way, the tangent ...
5
votes
1answer
319 views

If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?

I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant ...
3
votes
1answer
51 views

Involutive distributions?

How do we check exactly that a distribution is involutive? I have the following definition in my book: A $k-$dimensional distribution $\Delta$ on a manifold $M$ is a smooth choice of a k-dimensional ...
5
votes
0answers
58 views

Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
0
votes
0answers
14 views

Optimization on SE(3) with matrix logarithm

I am trying to optimize the following equation on manifold SE(3). $$Z(e^{\epsilon}) = \text{logm}{((e^{\epsilon}X)^{-1}W^{-1}e^{\epsilon}XY)}$$ Note that $W, X, Y, e^{\epsilon} \in SE(3)$ and $W, X, ...
3
votes
1answer
46 views

Non injective continuous maps

Motivated by comments on this question we ask the following question: Let $f:M\to M$ be a continuous map where $M$ is a compact manifold and $f$ is not injective. Are there necessarily ...
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1answer
54 views

topic between algebra and geometry [closed]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
2
votes
1answer
30 views

The restriction fo covering to a component is a covering map onto its image.

I am reading Lee's Introduction to Topological Manifolds. I got stuck on the problem 11-7 on pages 303. The below is the problem. Prove : If $q: E \rightarrow X$ is a covering map and $A \subseteq ...
0
votes
0answers
11 views

Transformation of a subset of compact Jordan sets to manifolds

Let $T$(for e.g. $[0,1]^2$) be a Jordan compact sets and $\tau$ be a "smooth enough one-to-one" transformation, i.e.($\tau: [0,1]^2 \rightarrow [0,1]^2 $). Lets take a subset of Lebesgue measure ...
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1answer
22 views

p-simplex spanned by elements in the boundary of the unit ball of Thurston norm is in the the boundary of this unit ball.

in completing my thesis I have reached a momentary impass. I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem to ...
2
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0answers
66 views

Understanding the “shape” of a singular Riemann surface

Consider the singular Riemann surface given by the following expression: $$z^d w^d-z^d-w^d+t=0\ ,$$ where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2. For $t\neq0$ the ...
0
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1answer
22 views

real coordinates of a complex manifold

I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one ...
2
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1answer
86 views

How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
1
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1answer
42 views

Basic examples topological manifolds with boundary

I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...
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0answers
37 views

n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
2
votes
3answers
49 views

Group actions on manifolds - exponential map

Let $M$ be a smooth manifold. Suppose $K$ is a Lie group (with Lie algebra $\mathfrak{k}$) acting EDIT: TRANSITIVELY on $M$ from the left and $G$ is a Lie group (with Lie algebra $\mathfrak{g}$) ...
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0answers
14 views

Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
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votes
0answers
20 views

Heegaard splitting via a Morse function - twisted union or not?

Let $M$ be a smooth, closed, connected, oriented 3-manifold and let $f: M \rightarrow \mathbb{R}$ be a self-indexing Morse function. Since $\frac{3}{2}$ is a regular value of $f$ it follows from Morse ...
0
votes
1answer
15 views

Critical and regular values of height functions on a closed hypersurface

Let $M$ be a closed connected hypersurface of $n$-dimensional in $\mathbb{R}^{n+1}=\{(x^1,\cdots,x^{n+1})\}$ and let $\nu$ be a smooth unit normal vector field of $M$ at $\mathbb{R}^{n+1}$, $H$ be the ...
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0answers
35 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
6
votes
1answer
87 views

Tangent space manifold

Let M be a differentiable manifold of dimension m and also let $\{\xi_1,\dots,\xi_m\}\subset \text{T}_pM$ be an linearly independent set of the tangent bundle of M at a certain point $p\in M$. I have ...
3
votes
2answers
53 views

Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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1answer
65 views

What's the geometrical meaning of immersion?

Does that just mean to different tangent vectors, their images are different tangent vectors?
1
vote
2answers
55 views

Does a proper map have to be continuous?

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...
3
votes
1answer
96 views

Looking for a good alternative to 'An introduction to manifolds' by Loring W. Tu

I'm currently studying some basic theory about manifolds from the book 'An introduction to manifolds' by Loring W. Tu. The problem I have with this book is that there are very little exercises, and ...
2
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1answer
59 views

Area form and surface area

I know how one can define the surface area via the charts of a surface in $\mathbb{R}^3.$ click here for instance Now, I read that the canonical surface area form for such a surface with surface ...
2
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1answer
54 views

Volume of Manifold with zero Lebesgue measure

Let $M$ be a smooth manifold in $\mathbb{R}^n$. If Lebesgue measure of $M$ is zero i.e $l(M)=0$, does it mean that volume of manifold is also zero i.e $Vol(M)=0$? Are they the same thing (volume and ...