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)

1
vote
0answers
27 views

Is there a relation between vectors on these two spaces?

I've been reading lately one paper on Physics, which basically presents one gauge theory approach to the problem of swimming at low Reynolds number. I've been trying lately to rewrite some of the ...
1
vote
0answers
17 views

Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
3
votes
1answer
27 views

Homology and Neighborhood

Let $X$ a connected manifold, $x \in X$ and $V$ a neighborhood of $x$. Assume $i:V \to X$ induce isomorphism between all homology groups. Does $X-p$ and $V-p$ still have the same homology groups ? ...
-1
votes
0answers
23 views

A manifold is a covering space over its quotient by a group action

Let $M \times G\to M$ be a properly discontinuous, free action of group $G$ on a manifold $M$. The quotient topology of the orbit space is Hausdorff. Suppose $p\in M$. How can we choose an open ...
5
votes
2answers
45 views

Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
2
votes
1answer
47 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
4
votes
0answers
46 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
3
votes
0answers
22 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
2
votes
0answers
27 views

Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. ...
2
votes
1answer
33 views

Compute in the chosen charts of $M$ and $S^1$ the expression of $DF_{(5,0,-4)}$

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5 x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. We ...
0
votes
1answer
25 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a ...
1
vote
1answer
47 views

Injectivity of the Differential of Smooth Map

I am trying to answer the following question: Let $M = \{(x,y)\in \mathbf{R}^2 : x^2 + y^2 < 1\}$. Define a smooth or $C^\infty$ function by $f\colon M \rightarrow \mathbf{R}^2$ as ...
4
votes
1answer
76 views

Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

So it is weekend! and I am reading a nice book, "The Poincaré conjecture", written by a mathematician (Donal O'Shea, topologist). The book introduces step by step basic concepts of Topology, and talks ...
2
votes
1answer
40 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
0
votes
1answer
21 views

$M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$

PROBLEM: $M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$. I know how to determine $T_pM$ and $N_pM$ for explicit examples, but I dont know how to handle this ...
0
votes
1answer
18 views

Product of $C^l$-Manifolds [closed]

Let $X$ and $Y$ be two $C^l$-Manifolds. Show that the cartesian product $X\times Y$ is a $C^l$-Manifold too. Can you guys give me a hint on how to show this?
3
votes
1answer
26 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
-3
votes
0answers
49 views

Show smoothness of this map

Let $S^3$ be the sphere identified with the subset $\mathbb{C}^2$ as $\{(x,y) \in \mathbb{C}^2; |x|^2+|y|^2=1\}.$ Then I want to show that the map $\phi: S^3 \rightarrow \hat{\mathbb{C}}$ is ...
1
vote
1answer
47 views

Open product neighbourhoods

Let $X\times Y$ be a topological space (if it helps anything, it can be assumed to be a smooth manifold and $U\subseteq X\times Y$ an open subset. Let $x\in X$ and $O\subseteq Y$ be open, such that ...
4
votes
1answer
70 views

Group action and smooth manifolds

I was wondering if it is for a compact (i.e. Hausdorff) smooth manifold $M$ sufficient to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
2
votes
0answers
64 views

Check Riemannian manifold's isometry to $\Bbb{R}^n$

Let $\mathcal{M}$ be the convex cone of symmetric positive definite $n\times n$ real matrices. $\mathcal{M}$ is an $\frac{n(n+1)}{2}$-dimenasional Riemannian manifold. Could you help me proving (or ...
2
votes
0answers
21 views

How to show that the inverse image of a face of a simple polytope is a connected manifold?

If $M^{2n}$ is a toric manifold over a simple polytope $P^n$ i.e; the orbit space of the action of the $(S^1)^n$ on $M^{2n}$ is an $n$ dimensional simple polytope $P^n$. Let $\pi : M^{2n} \rightarrow ...
2
votes
1answer
63 views

Is this one infinite dimensional manifold?

First of all, just to give context to the question: I've been reading some articles in Physics, and those articles imply without proof that one space is one infinite dimensional manifold. One of those ...
3
votes
0answers
55 views

$L^\infty$-bounds on eigenfunctions of Laplace-Beltrami opeator

Let $w_k$ be the eigenfunctions of the Laplace-Beltrami operator on a compact manifold $M$ without boundary. We assume that $\{w_k\}$ are orthonormal, thus $\|w_k \|_{L^2} = 1$. We know $w_k$ are ...
3
votes
0answers
60 views

Differentiating an endomorphism

Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote ...
0
votes
0answers
13 views

Two vector fields are cojugate but not take orbits

Let X and Y be C1 vector feilds on R^m. Suppose that 0 is an attracting hyperbolic singularity for X and Y. Show that there exists a homemorphism h of a neighborhood of origin which conjugate the ...
4
votes
1answer
55 views

Parametrizing the time an element stays in an open subset

Let $X$ be a topological space (If it helps anything, we can assume $X\subseteq\mathbb{R}^n$ or $X$ being a smooth manifold.) and $U\subseteq [0,1]\times X$ an open subset. Does there exist a ...
1
vote
0answers
15 views

Quickest way to restrict a homeomorphism

Let $\phi: U \to V \subset \mathbb{R}^n$ homeomorphism. My desire is: I want to say the restriction $\phi|_{\phi^{-1}(B_{r'}(x))}:\phi^{-1}(B_{r'}(x)) \to B_{r'}(x) $ is a homeomorphism in the ...
2
votes
0answers
13 views

Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
2
votes
3answers
35 views

2-Sphere surface coordinate dimension

Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...
1
vote
0answers
31 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
1
vote
1answer
36 views

Homeotopy of Shrinking Manifolds

Let $M$ be a $n$-dimensional open manifold in $\mathbb{R}^n$. Let $B^n_k$ be the closed $n$-dimensional ball of radius $k$. Let $$N_k = (M^c \oplus B^n_k)^c$$ where $X^c$ denotes the complement of ...
5
votes
1answer
124 views

Definition of a Cartesian coordinate system

Apologies if this is a basic question, but I'd really like to clarify the exact meaning of what a Cartesian coordinate system is. Heuristically, is it correct to say that a Cartesian coordinate system ...
0
votes
1answer
50 views

Smooth function, which separates between a closed and a open set.

Let $M$ be a smooth manifold, $O\subseteq M$ an open subset and $B\subseteq M$ a closed subset, such that $closure(O)\subseteq interior(B)$ I think there must exist a smooth function $f\colon ...
5
votes
1answer
87 views

Theorema egregium violated in dimension $n \ge 4$?

Gauß showed that for surfaces in $\mathbb{R}^3$ the Gaussian curvature ( = sectional curvature) is invariant under local isometries. This is known as the thema egregium. Now in another question ...
2
votes
1answer
33 views

Integration of $V$-valued differential form

When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as ...
1
vote
1answer
63 views

Meaning of “locally homeomorphic to $\mathbb{R}^{n}$”

I am fairly new to differential geometry and approaching it with a physics background (in the study of general relativity), as a result I'm having a few struggles with terminology etc, so please bear ...
1
vote
1answer
34 views

(Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture 1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere. 2.) ...
0
votes
0answers
13 views

Are variables in embedded space Statistically independent variables?

Performing Taken's phase space delay embedding on the observations $\mathbf{z}$ of a univariate random variable, with an embedding dimension $d$, we get a realization of $n$ points such as: ...
0
votes
0answers
45 views

Constructing coordinate maps on manifolds

I've been studying differential geometry for a little while now, but I've never properly justified to myself rigorously the need to consider other more general coordinate maps, other than Cartesian on ...
3
votes
3answers
37 views

Inverse function of $f(x,y,z) = (xy-z^2, x+z)$?

How do you determine the inverse function $f^{-1}: \mathbb{R}^2 \to \mathbb{R}^3$ of $f: \mathbb{R}^3 \to \mathbb{R}^2 , f(x,y,z) = (xy-z^2, x+z) $ ? Or to put it into a bigger context: ...
-1
votes
0answers
54 views

Show $L_{3,1}\sharp L_{3,1}$ and $L_{3,1}\sharp \overline{L_{3,1}}$ are not homotopy equivalent [closed]

Let $L_{p,q}$ with $(p,q)=1$ the usual Lens space, I must show that $L_{3,1}\sharp L_{3,1}$ and $L_{3,1}\sharp \overline{L_{3,1}}$ are not homotopy equivalent using homology/cohomology tools. Here, ...
1
vote
0answers
48 views

Equivalence of two norms on $L^p(M)$, $M$ compact manifold.

Let $(M,g)$ be a compact Riemannian manifold, $\mu(g)$ the Riemannian Lebesgue measure. Then we can define the usual $L^p$-spaces (lets assume $p<\infty$), $L^p(M,g):=L^p(M,\mu(g))$. For $f\in ...
0
votes
0answers
15 views

characeterization of zero sets of the riemannian measure of a riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold (does not have to be orientiable). Then there exists the Riemannian measure $\nu(g)$ on $M$. Let $(U_i,x_i)$ be a finite covering of $M$ of charts and let ...
5
votes
1answer
57 views

What are the 8 non-compact Euclidean 3-manifolds?

I have found several sources that mention that there are eight non-compact Euclidean 3-manifolds, with four orientable and four non-orientable. There are two standard references for this, but ...
6
votes
3answers
74 views

The set of all matrix with rank $n-1$ is a hypersurface.

Prove that the set $M$ of $n\times n$ matrices with rank $n-1$ is a hypersurface in $\mathbb{R}^{n²}$ and find the tangent space at $A=(a_{ij})$ where $a_{ij}=\begin{cases} \delta_{ij} \ \text{if} ...
2
votes
2answers
48 views

Prove that the antipodal mapping is an isometry on $S^n$. Help understanding the proof.

Prove that the antipodal mapping $A: S^n \to S^n$ given by $A(p)=-p$ is an isometry. I know that in order to prove that a map $f$ is an isometry of a smooth manifold $M$ it must hold true that ...
1
vote
1answer
32 views

Existence of a open set between a compact and an open set

Let $M$ be a compact manifold, $K\subset M$ compact, $U\subset M$ open. Does in this case always exist a open set $V\subset M$ such that $K\subset V\subset\bar{V}\subset U$ ?
1
vote
1answer
23 views

Lie Derivative of Connection 1 form

On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write \begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*} Here, $\omega$ is the ...
3
votes
1answer
74 views

Closed orientable 4-manifold with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1\times H^1\to H^2$

I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1(M;\Bbb Z_2)\times H^1(M;\Bbb Z_2)\to H^2(M;\Bbb Z_2)$. A non-orientable ...