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

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5
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2answers
206 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. ...
16
votes
2answers
301 views

Is this surface diffeomorphic to a 2-sphere?

Let $f:\mathbb{R}^3\to \mathbb{R}$ be defined by $f(x,y,z)=x^4+y^6+z^8$. Let $M=f^{−1}(1)$. Is $M$ is diffeomorphic to a sphere $S^2$? I tried to solve this problem, but I realized that I ...
6
votes
2answers
253 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?
3
votes
1answer
93 views

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$? I have the same question for ...
4
votes
1answer
190 views

Green's function for the Yamabe problem

I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8. Theorem 2.8 (Existence of the Green Function). Suppose $M$ is a ...
1
vote
1answer
28 views

Manifold with Negatives Identified

I have a three dimensional manifold where negatives are identified, so $x = -x$, but $x$ does not equal $cx$ unless $x$ is $1$ or $-1$. Does anyone know what this manifold is? Other than the ...
3
votes
1answer
274 views

Trivial tangent bundle and orientability

Let $M$ a (real) $n$-dimensional connected differentiable manifold. (a) The tangent bundle $TM$ is trivial, $TM \simeq M \times \mathbb R^n$; (b) $M$ is orientable. Consider the ...
2
votes
1answer
90 views

How to finish this proof (or sketch)?

I'm trying to prove that a manifold $M$, that is connected, is pathwise connected. I know the standard proof of this theorem: just use that the set of points that can be joined to a point $x \in M$ is ...
5
votes
5answers
403 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 ...
2
votes
1answer
97 views

Generalization of Grassmann manifold to include translations?

I came across a certain generalization of Grassmann manifolds and was wondering what work if any has been done on it. If you take the space of $n\times p$ real matrices, $n>p$, and define an ...
4
votes
2answers
140 views

If the pullback of $F:M \to N$ is an isomorphism, is $F$ a diffeomorphism?

Let $M,N$ be smooth manifolds, and let $F:M \to N$ be a smooth map. Then $F$ induces a map $F^*: C^\infty(N) \to C^\infty(M)$, given by $F^*(f)=f \circ F$. Suppose that $F^*$ is an isomorphism. Q1: ...
4
votes
2answers
248 views

Describe tangent and normal bundle to a manifold

Consider the set $X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$ I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
2
votes
1answer
95 views

Projective space is a manifold

I have no one else to correct my work since I am not going to school therefore I'd be very grateful if someone check my work. I was trying to show that the projective real plane is a manifold: On ...
0
votes
1answer
207 views

Concept and meaning of immersion

Who can explain concept and meaning of "Immersion" maps, very easy and useful? thanks for advanced.
0
votes
1answer
91 views

Relatively compact subsets of a manifold.

So I'm going through Otto Forster's "Lectures on Riemann Surfaces", and I need another hint (shame). This is in the "Cohomology Groups" sections, as part of a problem to show that for $X$ a compact ...
2
votes
1answer
78 views

Length of a curve on a manifold using diffeomorphisms

Lets say I have two (compact) manifolds $U$,$V$ and a diffeomorphism $\psi:U\rightarrow V $. The shortest way between two points $a$ , $b \in V$ is given by a parametrisation $\gamma :W \rightarrow ...
5
votes
3answers
250 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 ...
1
vote
0answers
40 views

Manifold with boundary : Definition using locally ringed space

Suppose we define a manifold with boundary, using the locally ringed space definition, with the local model being either open subsets of Euclidean spaces, or open subsets of the half-spaces in ...
3
votes
2answers
102 views

Is the image of a parametrization a manifold?

Consider this definition of the parametrization of a manifold, found in Hubbard & Hubbard: A parametrization of a $k$-dimensional manifold $M\subset\mathbb{R}^n$ is a mapping $\gamma:U\subset ...
8
votes
2answers
180 views

Structure of a $ C^{\infty} $-manifold

I was studying differentiable manifolds (an introduction) and found the following example, but I am confused. Example The function \begin{align} f: &\mathbb{R}^{3} \to \mathbb{R}, \\ f: ...
2
votes
0answers
83 views

Manifold definition using sheaves : Is the locally ringed condition necessary?

Let us suppose $X$ is a topological manifold, which we define as a Ringed Space with the local model being the Euclidean space $\mathbb R^n$ with the sheaf of continuous functions on it. Normally, ...
7
votes
0answers
89 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 ...
3
votes
1answer
54 views

vector fields in $\mathbb{R}^n$

I am seeing the notion of a vector field for the first time, and I am quite confused. The author (of Gauge Fields, Knots and Gravity) has defined a vector field to be any linear function $v: ...
1
vote
2answers
347 views

What is the topological classification of connected 1-manifolds? [duplicate]

Possible Duplicate: The only 1-manifolds are $\mathbb R$ and $S^1$ Any manifold is homeomorphic to the disjoint sum of its connected components. Therefore, the full classification of ...
4
votes
3answers
349 views

Topology on Klein bottle?

I was trying to show that the Klein bottle was second countable. My try was to use that it has the subspace topology of $\mathbb R^3$. Then I noticed that it is not imbeddable into $\mathbb R^3$. ...
1
vote
0answers
38 views

Approximation/Representation of local stable manifolds

I will give two preceding theorems and the question, which uses both, follows afterwards: Let $M$ be a smooth compact Riemannian manifold of dimension $n$ with a smooth measure $\mu$. $T_{x}M = ...
0
votes
0answers
84 views

Why should the tangent bundle of the boundary of a conctractible manifold be stably trivial?

the question is already clear from the title, but I have to add at least 30 useless characters. The question is equivalent to ask if the normal bundle of the boundary is stably trivial
0
votes
1answer
49 views

Determining whether a subset is a manifold given two different graphs, one of which is not everywhere differentiable

A smooth manifold in $\mathbb{R}^2$ is locally the graph of a $C^1$ function. Consider the graph of $f(x)=x^{1/3}$. Since $f$ is not differentiable at zero, we are in trouble. However, this subset is ...
5
votes
1answer
139 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
2
votes
1answer
181 views

Existence of bump functions which are positive on a prescribed set

Let $U \subset \mathbb{R}^n$ be an open subset of Euclidean space. I feel like there should be a smooth function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with $f|_U > 0$ and ...
1
vote
1answer
146 views

Transformation Rule for a Wedge Product of Covectors

Suppose two sets of covectors on a nector space $V$, $\beta^1,\ldots,\beta^k$ and $\gamma^1,\ldots,\gamma^k$, are related by $$\beta^i=\sum_{i=1}^ka^i_j\gamma^i,\quad i=1,…,k,$$ for a $k\times k$ ...
0
votes
1answer
40 views

Pairwise Disjoint Balls on a Manifold

I am wondering if it is always possible to find disjoint sets on any manifold such that these sets are balls when mapped to their locally Euclidean space $such$ $that$ there are an infinite number of ...
0
votes
0answers
47 views

$[0,1]\times [0,1]$ is a manifold with boundary [duplicate]

Possible Duplicate: Showing $[0,1] \times [0,1]$ is a manifold with boundary Definition: A manifold with boundary $M$ is a second countable Hausdorff space so that for a $p \in M$ there is ...
3
votes
0answers
81 views

Construct a space with free involution and homological restriction

I'm looking for a space $X$ which satisfies the following conditions: $X$ is a compact manifold. $H_\ast (X;\mathbb Z)$, the integral homology groups $X$, are torsion free. There is a free ...
4
votes
0answers
54 views

non-constant curve c with $\frac{\mathrm{d}c}{\mathrm{d}t}= \lambda\frac{\mathrm{d}^2c}{\mathrm{d}t^2}$

I don't know how to solve the following problem and would appreciate some help. Let $M$ be a submanifold of euclidean space and $c:[a,b] \to M$ a non-constant curve, such that the velocity field of ...
0
votes
1answer
142 views

How can I prove that “If $M$ is contractible differentiable manifold, then $M$ is orientable?”

If $M$ is a contractible differentiable manifold, then $M$ is orientable.
2
votes
1answer
84 views

Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations

Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$ Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on ...
1
vote
1answer
79 views

Submanifolds of a space of functions?

Let $f:\mathbb{R}\rightarrow(\mathbb{R}\rightarrow\mathbb{R})$ be a function mapping a real number uniquely into the set $\mathbb{F}$ of total functions from $\mathbb{R}$ to $\mathbb{R}$. $\mathbb{F}$ ...
2
votes
1answer
108 views

a theorem in topology

Is there anyone know there is a theorem in topology which states that a compact manifold "parallelizable" with N smooth independent vector fields. must be an N-torus? and why the vector field here is ...
4
votes
1answer
107 views

$U(n)/U(n-1)$ as homogeneous space

How can I prove that the quotient $U(n)/U(n-1) \simeq S^{2n+1}$ (where $U(n)$ is the unitary group). Is il correlated with the teory of homogeneous spaces?
3
votes
1answer
74 views

Is there any manifold that is not a subspace of a finite dimensional euclidean space?

I mean, is there any topological space that is locally euclidean, Haudorff and second countable and can't be embedded into a finite dimensional Euclidean space. I think it's hard for me to find such ...
13
votes
1answer
549 views

Showing $[0,1] \times [0,1]$ is a manifold with boundary

I'm familiarizing myself with manifolds. I tried to show $[0,1]\times[0,1]$ is a manifold with a boundary. Can you please tell me if my proof is correct: The definition for manifold with boundary: ...
2
votes
1answer
221 views

Why is this not a proof of Invariance of Domain?

We know that if $f:K \to X$ is continuous and injective, $K$ is compact, and $X$ is Hausdorff, then $f$ is a homeomorphism $K \cong f(K)$. So suppose $f:U \to \mathbb{R}^n$ is continuous and ...
1
vote
0answers
69 views

Velocity vectors on $S^{3}$

Consider $S^{3}$ as the unit sphere in $C^{2}$ under the usual identification $C^{2}\leftrightarrow R^{4}$. For each $z=(z^{1},z^{2})\in S^{3}$, define a curve $\gamma _{z}:R\rightarrow S^{3}$ by ...
1
vote
1answer
51 views

Question on Notation

In Loring Tu's text An Introduction to Manifolds, Exercise $2.4$ asks us to show that $D_1\circ D_2$ need not be a derivation while $D_1\circ D_2-D_2\circ D_1$ is always a derivation. My question is ...
2
votes
2answers
149 views

The most general notion of a directional derivative

Questions: I know you can define a directional derivative on some subset of $\mathbb R^n$, but what can be said about an arbitrary set of points, $S$? What are the most general criteria $S$ must ...
1
vote
2answers
159 views

The dimension of linear map

I am reading "Introduction to smooth manifolds" by Lee and one place is very unclear for me: Let $P$ and $Q$ be any complementary subspaces of $V$ (which is an $n$-dimensional real vector space) of ...
2
votes
2answers
123 views

Gluing cylinders together

I was trying to glue two cylinders and then show that the resulting space is a manifold. Here is the first of my attempts: The cylinders I denote $C_0 = S^1 \times [0,1]$ and $C_1 = S^1 \times ...
5
votes
0answers
123 views

Gluing manifolds

Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this: A point $(\cos \phi , \sin \phi, ...
2
votes
1answer
73 views

Neighborhoods of half plane

Define $H^n = \{(x_1, \dots, x_n)\in \mathbb R^n : x_n \ge 0\}$, $\partial H^n = \{(x_1, \dots, x_{n-1},0) : x_i \in \mathbb R\}$. $\partial H^n$ is a manifold of dimension $n-1$: As a subspace of ...