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

learn more… | top users | synonyms (1)

7
votes
3answers
505 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
7
votes
2answers
176 views

How do manifolds have enough structure to do calculus?

I am referring, of course, to to differentiable manifolds. I've seen a few different definitions. The one I like best is the one which says it's a topological space such that every point has a ...
7
votes
2answers
494 views

Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

Let $(M,g)$ be a Riemannian manifold. Then I want to show that these are equivalent: (i) Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal. ...
7
votes
2answers
707 views

A Banach Manifold with a Riemannian Metric?

Given an infinite dimensional manifold modeled on a Banach space, what does it mean for it to have a Riemannian metric? Does it necessarily mean that it is actually a Hilbert manifold? My ...
7
votes
1answer
274 views

Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi $ sends $ U $ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I ...
7
votes
3answers
414 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
7
votes
3answers
741 views

Definition of manifold

From Wikipedia: The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. A topological manifold is a topological ...
7
votes
1answer
96 views

What kind of tools do we have to detect when a manifold is a product of other manifolds?

What sort of tools are out there that can detect when a manifold is a product of other manifolds? For example, comparing the homology of the circle to the torus, the homology of the torus gets more ...
7
votes
3answers
98 views

Embeddings (how to prove them exactly)

For which of the following sets is the statement: '$A$ can be embedded in $B$' true? I can try to decide this intuitively but don't know if I'm right, and surely don't know how to formally prove it. ...
7
votes
2answers
238 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
7
votes
1answer
461 views

A Cover of an Orientable Manifold is Orientable

The following question comes from Introduction to Smooth Manifolds by Lee: Suppose $\widetilde{M}$ smoothly covers $M$ where $M$ is orientable. Show that $\widetilde{M}$ is orientable. I think the ...
7
votes
2answers
72 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
7
votes
2answers
87 views

Detecting compactness from the ring of smooth functions

Given a smooth manifold $M$, is there some ring-theoretic property (preferably not mentioning $M$) such that $C^{\infty}(M)$ has this property if and only if $M$ is compact?
7
votes
1answer
475 views

Complement of figure-8 knot

I am reading W. Thurston's famous "3-dimensional Geometry and Topology", but I am stuck at the point where it is said that gluing two tetrahedra in an appropriate way give you the complement of the ...
7
votes
1answer
163 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 ...
7
votes
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 ...
7
votes
1answer
1k views

Isometries preserve geodesics

Let $f$ be an isometry (i.e a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds $(M,g)$ and $(N,h).$ One can argue that $f$ also preserves the induced metrics $d_1, ...
7
votes
1answer
142 views

There is no immersion of the Möbius band in the plane.

There is no immersion of the Möbius band in the plane. I believe we have to work with the tangent bundle of the Möbius band, but I'm not getting no useful result.
7
votes
2answers
191 views

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
votes
1answer
809 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 ...
7
votes
2answers
138 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
votes
3answers
485 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: ...
7
votes
1answer
378 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
votes
2answers
95 views

What are some good sources to learn about real analytic manifolds?

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade ...
7
votes
1answer
103 views

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable (Tensors, or k-linear forms)

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable. $f\in A_k(V)$ is decomposable if there exists a $a_1,...,a_k\in V^\wedge$ such that $f=a_1\wedge...\wedge a_k$ In this case "let ...
7
votes
1answer
132 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
votes
1answer
153 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
votes
1answer
211 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
votes
0answers
69 views

Is a compact, simply-connected 3-manifold necessarily $S^3$ with $B^3$'s removed?

Let $M$ be a compact, simply-connected 3-manifold (which is also smooth and connected). Is $M$ diffeomorphic to $S^3$ with a finite number of $B^3$'s removed? This seems like a handy fact, but I ...
7
votes
0answers
161 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
votes
0answers
138 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
votes
2answers
338 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 ...
6
votes
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
votes
2answers
595 views

Any example of manifold without global trivialization of tangent bundle

It is said for most manifolds, there does not exist a global trivialization of the tangent bundle. I am not quite clear about it. The tangent bundle is defined as $$TM=\bigsqcup_{p\in M}T_PM$$ So is ...
6
votes
2answers
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
votes
2answers
325 views

Proof of whitney's embedding theorem?

While learning about the rigorous definition of manifolds, my text mentions that any $n$-dimensional manifold can be embedded in $\Bbb{R}^{2n}$, which is called Whitney's embedding theorem. I have ...
6
votes
2answers
433 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 ...
6
votes
2answers
294 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
votes
1answer
463 views

Are there p-adic manifolds?

Is there anything resembling a manifold on the field of p-adic or complex p-adic fields? If so is there a connection to algebraic geometry as rich as in the reals?
6
votes
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
votes
5answers
195 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
votes
2answers
247 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
votes
1answer
618 views

How to prove figure eight is not a manifold? [duplicate]

Possible Duplicate: A wedge sum of circles without the gluing point is not path connected I know that figure eight is not a manifold because its center has no neighborhood homeomorphic to ...
6
votes
1answer
124 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
votes
2answers
227 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
votes
1answer
196 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
votes
1answer
272 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
votes
2answers
799 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
votes
1answer
167 views

$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of ...
6
votes
1answer
114 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?