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

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Do we know this homogeneous space by another name?

Consider the homogeneous space $GL(3)/GL(2) = GL(3,\mathbb{R})/GL(2,\mathbb{R})$ where $GL(2)$ fixes the first coordinate axis (so can be identified with the subgroup of $2\times 2$ blocks sitting in ...
7
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2answers
128 views

Finding a subspace whose intersections with other subpaces are trivial.

On p.24 of the John M. Lee's Introduction to Smooth Manifolds (2nd ed.), he constructs the smooth structure of the Grassmannian. And when he tries to show Hausdorff condition, he says that for any 2 ...
7
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1answer
346 views

De Rham cohomology of $\mathbb{RP}^{n}$

Consider map from $S^{n}$ to $\mathbb{RP}^{n}$ $$\varphi:S^{n}\to\mathbb{RP}^{n}$$ which maps point $x\in S^{n}$ to corresponding direction in $\mathbb{R}^{n+1}$. This map induces map ...
7
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1answer
146 views

What is the relationship between Grassmann Manifolds with different dimensions?

I'm an EE student and I'm just beginning to learn about the Grassmann Manifold. As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector ...
7
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1answer
194 views

What's the intuition behind the tangent bundle?

Well, when we work with a smooth manifold $M$ we can associate with each point $p\in M$ a vector space $T_p M$ of all vectors at $p$ tangent to $M$: this is the space of linear functionals obeying ...
7
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1answer
440 views

Is a connected sum of manifolds uniquely defined?

It is a standard excercise in differential geometry to prove that a connected sum $M\#N$ of two smooth manifolds $M,N$ of the same dimension is uniquely defined (under some assumptions regarding ...
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0answers
116 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
7
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0answers
249 views

Munkres' Question on Manifolds

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads: QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points ...
7
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1answer
70 views

Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^n$. Can every question about ...
7
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0answers
113 views

Translating a passage of a paper by L. Bérard Bergery

I am currently studying the following paper on Einstein manifolds: L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Elie Cartan, Univ. Nancy №6, 1-60 (1983). I have ...
7
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95 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
7
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2answers
312 views

Seifert matrices and Arf invariant — Cinquefoil knot

I have computed the following Seifert matrix for the Cinquefoil knot: $$ S = \begin{pmatrix} 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 \\  0 & 1 & 1 & 0 \\ 0 ...
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6answers
2k views

What math is necessary to learn manifolds?

I want to learn about manifolds, but I'm only a senior in high school and obviously have a while to go. I'm in AP Calc BC. What should I study to eventually learn manifolds? Linear Algebra? What ...
6
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2answers
396 views

Question about definition of topological manifold

The following definition of topological manifold is given in Lee's Introduction to topological manifolds (2000) on page 33: A topological manifold is a second countable Hausdorff space that is ...
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1answer
1k views

Interior and boundary points of $n$-manifold with boundary

I'm reading Lee, 'Introduction to Topological Manifolds', 2011. After he introduces $n$-manifold with a boundary An $n$-manifold with a boundary is a second countable Hausdorff space in which any ...
6
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2answers
255 views

Orientable double covers for non-orientable manifolds

If I have two non-orientable connected manifolds such that their orientable double covers are homeomorphic, can anything be said about the manifolds? Are they homeomorphic?
6
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2answers
45 views

Frobenius theorem for 2-plane fields on some open set in $\mathbb{R}^3$

I need help with this two part question. I am rather confused by it. let $f(x, y, z)$, $g(x, y, z)$ be smooth on $U \subset \mathbb{R}^3$ with $f^2 + g^2 > 0$ on $U$. Define the differential form ...
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2answers
1k views

How to show that the unit sphere is a topological manifold?

Sorry for this basic question, but I´m not sure of something. I want to see one example. The definition of a n-manifold is a Hausdorff space, such that each point has an open neighborhood homeomorphic ...
6
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2answers
287 views

The Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a $3$-manifold which may be viewed as a circle bundle over ...
6
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5answers
172 views

How would one define a “manifold” object in prose writing?

I have a question that I fear may raise some objection to the fact that it has been posted here, but I cannot think of a more appropriate place to pose it. I am not a mathematician; I'm a historian, ...
6
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2answers
243 views

This set is a manifold

let $S$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show this is a topological manifold. for starters one needs to define a suitable topology on ...
6
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2answers
704 views

Manifold with different differential structure but diffeomorphic

I'm new to differential geometry and reading Lee's book Manifold and Differential Geometry. In the first chapter, he mentioned the following two maps on $\mathbb{R}^n$: (1) $id: (x_1,x_2\cdots x_n) ...
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1answer
110 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. ...
6
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2answers
167 views

Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
6
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1answer
173 views

Difference between two definitions of Manifold

I've been studying Differential Geometry on Spivak's Differential Geometry book. Since Spivak just works with notions of metric spaces and analysis, I'm doing fine. The point is that Spivak presents ...
6
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1answer
239 views

Uniqueness of Smoothed Corners

Let $M^m$ be a smooth manifold with boundary. We may form a new topological manifold by "adjoining a handle to $M$", I.e. by choosing a smooth map $h : \partial \mathbb{D}^{\mu} \times ...
6
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2answers
615 views

$\mathbb{C}\mathbb{P}^1$ is homeomorphic to $S^2$

I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In ...
6
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1answer
95 views

Is there any embedding theorem for fibre bundles?

I would like to know whether there is an embedding theorem for fibre bundles, like Whitney embedding theorem. When can a given fibre bundle be a subbundle of some higher dimensional bundle?
6
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2answers
252 views

Why this topological space is not a topological manifold?

I'm having troubles to prove that the following space is not a topological manifold: Let $r:S^1\to S^1$ be a rotation of $\frac{2\pi}{3}$, i. e., ...
6
votes
2answers
138 views

homomorphisms of $C^{\infty}(\mathbb R^{n})$

Let $F: \mathbb R^{n} \to \mathbb R^{m} $ be a smooth map, then we have homomorphism of algebras $F^{*}: C^{\infty}(\mathbb R^{m}) \to C^{\infty}(\mathbb R^{n})$. Is it true that any homomorphism of ...
6
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1answer
87 views

If compact simply connected manifold has the same rational homotopy groups as $S^n$ or $\mathbb{C}P^n$, must it have the same cohomology ring?

The question came up while trying to shorten a paper I'm writing into submission-ready length. Let $M$ be a compact simply connected manifold. By defininition, the rational homotopy groups of $M$ ...
6
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2answers
255 views

Ideal of smooth function on a manifold vanishing at a point

I'm trying to prove the following lemma: let $M$ be a smooth manifold and consider the algebra $C^{\infty}(M)$ of smooth functions $f\colon M \to \mathbb{R}$. Given $x_0 \in M$, consider the ideals ...
6
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2answers
173 views

Topological space M with partition of unity--->M paracompact. John Lee Problems

Suppose $M$ is a topological space with the property that for every open cover $X$ of $M$, there exists a partition of unity subordinate to $X$. Show that $M$ is paracompact.
6
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1answer
202 views

Orientability of Manifolds

Given that $f \colon \mathbb R^n \rightarrow \mathbb R$ is a smooth function and if $c \in \mathbb R$ is a regular value how would I go about showing that $f^{-1} (c)$ is an orientable manifold? ...
6
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1answer
142 views

Looking for an atlas with 1 chart

Can we provide the set $\{(x,y,z)\in\mathbb{R^3}|x^2+y^2=1\}$ with a 2-dimensional manifold structure involving only 1 chart? I can see it with 2 charts with cylindrical coordinates, but not with only ...
6
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2answers
121 views

How does smoothness prevent “singularities”?

This is a refinement of one of my earlier questions (I failed to put into words what I really wanted to ask). First of all, I'm not sure "singularity" is the correct word to use hence the quotes. ...
6
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2answers
505 views

Is a covering space of a manifold always a manifold

Assume $M$ is a manifold and $q : E \to M$ is a covering map. I have been told a few times that a covering space of a manifold is again a manifold. Indeed, it is easy to verify that $E$ is both ...
6
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1answer
2k views

The special orthogonal group is a manifold

How can we show that $SO(n)$ is an $n^2$-manifold. It would be tempting to say that $SO(n)$ is an open set of $\mathbb R^{n^2}$ but this is not the case since $SO(n)$ is given as the intersection of ...
6
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1answer
111 views

Is this a manifold?

I am trying to get started with differential geometry, and am having a difficult time wrapping my head around the concept of a manifold. One thing that would make it easier to understand would be if ...
6
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1answer
95 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
6
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1answer
170 views

Proving that Hermitian Metric yields Hermitian Structure on Complex Manifold

Let $g$ be a Riemannian metric on an almost complex manifold $(M,J)$. Suppose $g$ is Hermitian in the sense that $$g(JX,JY) = g(X,Y)$$ Let $\Omega$ be the associated fundamental (Kahler) form ...
6
votes
1answer
290 views

Uniformization Theorem for compact surface

Why in proof of proposition 6 of http://arxiv.org/abs/0909.1665, they claim that if a embedded surfaces $\Sigma^2 \subset (M^3,g)$ is homeomorphic to $\mathbb{RP}^2$, where $M$ is compact manifold, ...
6
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2answers
926 views

Why are Riemann surfaces algebraic curves?

I've never fully understood the connection between Riemann surfaces and algebraic varieties. I'm particularly interested in the case of the modular curve of level N--I know how the Riemann surface is ...
6
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1answer
769 views

Converse To Quotient Manifold Theorem [Exercise in Lee Smooth Manifolds]

I would like help with the following problem (chapter 9, #4) from Lee's Smooth Manifolds [its not homework, I'm reading it and I got stuck on this one] If a Lie group $G$ acts smoothly and freely on ...
6
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1answer
69 views

If it looks like a solid torus, walks like a solid torus, and quacks like a solid torus, is it a solid torus?

If an orientable 3-manifold $M$ has boundary the torus $S^1\times S^1$ and deform retracts to a solid torus $S^1\times D^2$, is it necessarily homeomorphic to a solid torus? Equivalently, if the ...
6
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1answer
99 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
6
votes
1answer
116 views

Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
6
votes
1answer
137 views

Is there any relationship between Cauchy-Riemann equations and vector fields on manifolds?

Well, suppose we have $f : \mathbb{C} \to \mathbb{C}$ analytic, then if $f = u + iv$ the functions $u,v : \mathbb{C} \to \mathbb{R}$ satisfy the Cauchy-Riemann equations: $D_1u=D_2v$ and $D_2u=-D_1v$. ...
6
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1answer
224 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
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3answers
439 views

Lie algebra action from Lie group action: coordinates

Here's the setup: I have $SL(2;\mathbb{C})$ acting on $V = \mathbb{C}[z,w] = \oplus_d V_d$, where $V_d$ is the homogeneous complex polynomials of degree $d$. The action is precomposition: ...