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)

5
votes
4answers
259 views

Transition functions in manifold

Following is given as the definition of a manifold in a book which I am reading. Let $\{U_{\alpha}:\alpha\in I\}$ be an open covering of a Hausdorff topological space $X$ and $\phi_{\alpha}$ be ...
5
votes
4answers
186 views

Is $[0,1]$ a 1-manifold?

Is $[0,1]$ a 1-manifold? I would say no because at either endpoint the open sets containing it aren't homeomorphic to a 1-ball in $\mathbb R^1$.
5
votes
2answers
2k views

Manifold Explained

Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one. My ...
5
votes
2answers
600 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
5
votes
2answers
147 views

What is $S^3/\Gamma$?

Let G is a group and H is a subgroup of G. I know $G/H$ is the quotient space but I have no idea about what $S^3/\Gamma$ is, where $S^3$ is the sphere and $\Gamma$ is a finite subgroup of $SO(4)$. In ...
5
votes
2answers
573 views

Manifolds and Charts

I have a very silly and basic question about finding charts for a manifold. The point is: I'm self learning differential geometry, however, I didn't find the answer for this in the book nor on the ...
5
votes
3answers
575 views

Geodesic on a plane

I guess that each geodesic on a plane is a straight line. Is it right? What can I use to prove it? I guess I have to use somehow Levi-Civita connection.
5
votes
3answers
177 views

Kernel of the Laplacian on a compact manifold

Is there a way to characterise the kernel of the Laplace-Beltrami operator on a compact manifold without boundary? Or is it just "the set of functions $u$ such that $-\Delta u = 0$?"
5
votes
2answers
187 views

Why is $\partial\partial M=\varnothing$?

Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
5
votes
3answers
790 views

Questions about definition of Tangent Space

I have some questions about the definition of tangent space that arose after reading the book Differential Geometry of Curves and Surfaces of Manfredo do Carmo. First, what's the best way to define ...
5
votes
1answer
298 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
5
votes
1answer
798 views

Manifold embedded in euclidean space with nontrivial normal bundle

Let $X$ be a differentiable $n$-manifold embedded in some $\mathbb{R}^{n+1}$. I have two questions. I have read that if $X$ is compact and orientable, then the normal bundle of the embedding is ...
5
votes
1answer
100 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
5
votes
1answer
68 views

Concerning the tangent space of an exotic $\mathbb R^4$

My geometric intuition is very poor, so my naive approach to this question is "if $M$ is an exotic $\mathbb R^4$, then $TM$ must be something like $\mathbb R^8$, which is not exotic". Of course, my ...
5
votes
1answer
610 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 ...
5
votes
2answers
976 views

Spivak's proof of Inverse Function Theorem

I am having trouble with Spivak's proof of the Inverse Function Theorem in his Calculus on Manifolds: 2-11 Theorem (Inverse Function Theorem). Suppose that $f: \mathbb{R}^n\to\mathbb{R}^n$ is ...
5
votes
2answers
68 views

Cut-number of Klein bottle and other non-orientable surfaces

What is the maximum number $c$ (cut-number) of non-intersecting (edit: two-sided) circles on a Klein bottle $N_2$ and, in general, a surface $N_h$ with $h$ Möbius strips, such that cutting by these ...
5
votes
5answers
635 views

Different definitions of a “one-form”

I started self-studying some differential geometry while using several different sources, but I'm confused about the notion of a one-form and how different places define it differently. Here are some ...
5
votes
3answers
340 views

Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$

I am trying to construct a diffeomorphism between $S^1 = \{x^2 + y^2 = 1; x,y \in \mathbb{R}\}$ with subspace topology and $\mathbb{R P}^1 = \{[x,y]: x,y \in \mathbb{R}; x \vee y \not = 0 \}$ with ...
5
votes
2answers
551 views

orthonormal vector fields

In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$? It seems ...
5
votes
3answers
116 views

How much does $\operatorname{Aut}(H_1(S))$ determine a homeomorphism $S \to S$?

Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$. How much can we say the converse? Namely, if we are given an element of ...
5
votes
1answer
304 views

Can any smooth manifold be realized as the zero set of some polynomials?

Is any real smooth manifold diffeomorphic to a real affine algebraic variety? (I.e. is there an "algebraic" Whitney embedding theorem?) And are all possible ways of realizing a manifold $M$ as an ...
5
votes
1answer
165 views

Different definitions of differential forms?

I am a physicist and was reading about differential forms in Classical Mechanics. Now, I thought that a two-form is a smooth map $\omega : M \rightarrow \Lambda(T^*M)$ so that a point $p$ on the ...
5
votes
2answers
124 views

frustrating experience about differential geometry

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress... Indeed my major is mainly numerical analysis. I am studying modern geometry, ...
5
votes
1answer
172 views

Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way?

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth ...
5
votes
2answers
658 views

First proof of Poincaré Lemma

I know that a way of proving Poincare lemma is to use the homotopy invariance and contractibility of the Euclidean space. Is there is a way of doing it directly (without using the contractibility of ...
5
votes
2answers
168 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 ...
5
votes
1answer
202 views

Is the configuration space a manifold? a CW complex?

The ordered configuration space of $n$ points in a topological space $X$ is defined as $F(X,n)=\{(x_1,\ldots,x_n)\in X^{n} | x_i\neq x_j \mbox{ for } i\neq j\}$ and the unordered configuration space ...
5
votes
1answer
393 views

$M/\Gamma$ is orientable iff the elements of $\Gamma$ are orientation-preserving

The probem is: Suppose $M$ is connected, orientable, smooth manifold and $\Gamma$ is a discrete group acting smoothly, freely, and properly on $M$. We say that the action is ...
5
votes
1answer
110 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
5
votes
1answer
106 views

Is there any way to define differentiablity without any reference to the Euclidean space?

We define metric spaces based on the properties of the real numbers $\Bbb{R}$. In the same spirit we define smooth manifolds. But there is a more general and elegant way to formulate our intuition of ...
5
votes
1answer
177 views

Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?

I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds: Example 9.3. Let $\Gamma$ be the graph of the function $f(x) = \sin(1/x)$ on the ...
5
votes
2answers
876 views

Relationship Between Differential Forms and Vector Fields

I am trying reach an understanding of precisely how the space of differential forms is related to the space of vector fields. These are the definitions that I understand and am using for these ...
5
votes
2answers
175 views

Classical tensor analysis and Tensors on Manifolds

I learned tensors the bad way (Cartesian first, then curvilinear coordinate systems assuming a Euclidean background) and realize that I am in very bad shape trying to (finally) learn tensors on ...
5
votes
2answers
307 views

Why do folded concentric circles and rectangles form a hyperbolic paraboloid?

Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?
5
votes
2answers
287 views

Compatibility of a connection and metric

Every Riemannian manifold admits a metric connection. Suppose $M $ is a manifold and $\nabla $ is an arbitrary connection on the tangent bundle. Does $M$ necessarily admit a metric such that $\nabla$ ...
5
votes
1answer
304 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
5
votes
1answer
188 views

Is a continuous map between smoothable manifolds always smoothable?

Let $X$ and $Y$ be topological manifolds and $f:X\to Y$ a continuous map. Suppose $X$ and $Y$ admit a differentiable structure (at least one). My question: is it always possible to choose a ...
5
votes
1answer
289 views

Hopf Fibration in Local Coordinates

I have the following task: Consider the unit sphere $\mathbb S^3$ in $\mathbb R^4$. We know $\mathbb CP^1\simeq \mathbb S^2$ (homeomorphic). Identifying $\mathbb R^4$ with $\mathbb C^2$, we have a ...
5
votes
2answers
674 views

Question about Lee's Introduction to Topological Manifolds

From page 2 in Lee's Introduction to topological manifolds: Question 1: What does "describe parametrically" exactly mean? Is it a synonym for "global coordinate chart"? (that is, an atlas ...
5
votes
1answer
250 views

Question about $4$-manifolds and intersection forms

This is a question related to an earlier question of mine: I've been reading about topological invariants. Some of them are defined in terms of quadratic forms. My current understanding is: we can ...
5
votes
2answers
299 views

A question about the topological homogeneity of manifolds

Let $M$ be a connected topological manifold of dimension $n \geq 1$. It is well known that if $p,q \in M$ then there exists a homeomorphism $\phi : M \to M$ such that $\phi(p) = q$. This can be summed ...
5
votes
2answers
549 views

Which mapping tori are Seifert manifolds?

According to Orlik's lecture notes on Seifert manifolds (and the Wikipedia page on Seifert fiber spaces), a mapping torus over a 2-torus is a Seifert manifold if and only if it is the mapping torus of ...
5
votes
1answer
175 views

How to find the integral curves that are orbits of one-parameter groups?

Consider $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta ...
5
votes
1answer
116 views

Interpretation of generalized eigenvector in orbits

First of all, this is my fourth question about dynamical systems in a week, sorry for that. Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase ...
5
votes
1answer
94 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
5
votes
1answer
65 views

Proof of 'manifold with dimension less then 4 always has differentiable structure'

L.S., I read in the lecture notes of my course on manifolds (undergraduate) a little side-note that stated that every manifold with dimension less then 4 can be equipped with a differentiable ...
5
votes
2answers
238 views

Injectivity of a map between manifolds

I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ ...
5
votes
2answers
1k views

Orientability of projective space

Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even. First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with ...
5
votes
2answers
253 views

Are matrices with determinant zero a manifold?

Consider the set of matrices with determinant zero in $M_n(\mathbb R)$, where $n > 1$. Is it a manifold? In fact, is it even a topological manifold? I would suspect not; but I do not have a proof. ...