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

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2
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1answer
26 views

Expression of a given vector field for the stereographic projection of the sphere

I have got stuck trying to solve the following problem. Let $X=-zx \frac{\partial}{\partial x} -zy \frac{\partial}{\partial y} + (1-z^2) \frac{\partial}{\partial z}$ be a vector field in ...
0
votes
2answers
31 views

Does the forgetful functor from smooth manifolds to sets preserve colimits?

It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits? It certainly preserves all examples of ...
1
vote
1answer
53 views

Calculating Volume of surface of unit sphere

I am trying to understand the proof for $w_n = 2\pi^{n/2}/\Gamma(n/2)$ where $w_n$ is the volume of the surface $S_n$ of the n-dimensional unit sphere $K_n$. There is stated that $Vol(K_n) = ...
0
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0answers
18 views

Grassmannian Non-Convex

The Grassmannian manifold $Gr(r,V)$ defines the set of $r$-dimensional linear subspaces of the vector space $V$. My question is, in general, what is the simplest way to see that $Gr(r,V)$ is a ...
1
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2answers
73 views

Rank of Jacobian Matrix for the Stereographic Projection

With the definition $S^{n} = \{\ \mathbf{x} \in \mathbb{R}^{n+1}\ | \ ||\mathbf{x}|| = 1\ \}$, and the function $\ f:\mathbb{R}^{n} \to S^{n} \setminus \{ (0,...,0,1) \}$ defined by: $f(\mathbf{u}) ...
9
votes
0answers
151 views

Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an ...
2
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0answers
45 views

Manifolds or Complex Analysis for Algebraic Geometry? [closed]

I'm an undergraduate and I have one year left to take some courses at the graduate level to prepare myself for graduate school. I go to a quarter school (U. Washington) so I only have time to take two ...
0
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0answers
39 views

$C^k$-maps between manifolds is a sheaf?

I know that the functor from the category of open subsets of a manifold $M$ to the Sets, taking an open set $U$ and associating to it the collection of $C^k$ maps to $\mathbb{R}$ is a sheaf. My ...
2
votes
0answers
33 views

Wedge product of $k$-forms

I'm studying smooth manifolds with Lee's book. He defines a $k$-form on a manifold $M$ as a section $M \to \Lambda^k M$ (where $\Lambda^k M = \bigsqcup_{p\in M} \Lambda^k T_pM$ is the smooth vector ...
10
votes
1answer
182 views

Is It Always Possible to Cross a Surface Exactly Once?

Yesterday, in my physics class, the following question arose: Is there a closed surface embedded in $\mathbb R^3$ dividing space into two connected components such that all paths from one ...
0
votes
1answer
40 views

Is $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$ a manifold of class $C^{\infty}$?

Let $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$. Is $M$ a manifold of class $C^{\infty}$? I need find a atlas $\{(U_i,\varphi_i)\}_{i\in I}$ with $U_i$ open sets and $\varphi$ ...
1
vote
0answers
53 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...
0
votes
2answers
21 views

Brouwer degree extension Lemma

Let $M$ and $N$ be oriented $n$-dimensional manifolds without boundary an also $M$ is compact and $N$ connected. Suppose that $M$ is the boundary of a compact oriented manifold $X$ and that $M$ is ...
2
votes
2answers
85 views

Are $\mathbb{C}^2$ and $\mathbb{C}^2/(x,y)\sim(y,x)$ homeomorphic?

Let $A$ be the set of monic quadratics over $\mathbb C$ and let $B$ be the set of unordered pairs over $\mathbb C$ where possibly the two elements of the pair may be the same. Then the map which takes ...
2
votes
0answers
23 views

Connection on $T\mathbb{R}^n$

Let $\nabla$ be a connection on the tangent bundle $T\mathbb{R}^n$. Now, I need to show that there exist smooth function $C_i: \mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$, $i=1,\dots ,n$ such ...
0
votes
0answers
21 views

Direct Sum Decomposition and Multiplicity of Eigenvalue Zero

I am currently reading a paper which states the following: Let be given a function $f(x):\mathbb{R}^n \rightarrow\mathbb{R}^n$ with its zero set $\mathcal{V}(f)$. If for every $a\in\mathcal{V}(f)$ ...
1
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0answers
33 views

Proving smooth map between smooth manifolds is constant based on push forward being zero

I have just me this problem in my class on smooth manifolds from Lee's introduction to smooth manifolds, from the chapter on the tangent bundle stating the following: Let M, N be smooth manifolds, ...
1
vote
1answer
25 views

Completeness of “weighted” shortest path metric

I am trying to see when this type of metric is complete: Let $A$ be the set of $C^{1}$ paths in $U \in \mathbb{R}^{n}$. For any $x,y$ define $$\rho(x,y) = \inf_{\gamma \in A; \gamma(0) = x, \gamma(1) ...
-2
votes
1answer
43 views

A function $\phi$ between two manifolds of class $C^\infty$ is constant if $d\phi$ [closed]

Let $M$ and $N$ two manifolds of $C^{\infty}$, $M$ connected, and $\phi:M\to N$ also of class $C^{\infty}$ so that in all point $m\in M$ the function $d\phi(m):M_m\to N_{\phi(m)}$ between the ...
0
votes
0answers
42 views

Brouwer Degree is locally constant

I'm reading Milnor's book "Topology from the differential viewpoint" and I'm stuck at this point: Let $M$ and $N$ be oriented n-dimensional manifolds without boundary and let $$f: M \longrightarrow ...
1
vote
1answer
50 views

Stokes theorem for Cuboid

I need to proof stokes theorem $\int_Qd\omega=\int_{\partial Q}\omega\;$ for a 2-form and $Q\subset \mathbb R^3 \;$a cuboid. Since $\omega \;$ is a two form it can be written as $$\omega ...
4
votes
2answers
66 views

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$?

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$? For starters, I know that if the $n$-dimensional $M$ can be immersed in ...
0
votes
1answer
29 views

Do Killing vector fields satisfy $\nabla_a X^a + \nabla_b X^b=0$?

Killing vector fields are those that verify $\mathcal{L}_X (g)=0$. This is equivalent to the following equation for a coordinate basis: $$\nabla_a X_b + \nabla_b X_a=0$$ Do Killing vector fields ...
0
votes
0answers
12 views

constructions over rotation surfaces in $\mathbb{R}^3$

Let $f: (0, \infty) \times \mathbb{R} \to \mathbb{R}$ be continuously differentiable, and $\nabla f(x) ≠ 0$ on the set $M = f^{-1}(0)$. (Which means that $M$ is a 1-dimensional manifold of the ...
5
votes
1answer
80 views

Find a (simple?) counterexample to this statement about topological manifolds.

Let us assume by a topological manifold $M$ of dimension $n$ I mean a Hausdorff topological space that is locally homeomorphic to $\mathbf{R}^n$, where $n$ is fixed. I know that if $M$ is assumed ...
2
votes
1answer
42 views

How is this “Stiefel manifold”-like fiber bundle called? How to characterize it?

Let $M$ be a smooth manifold of dimension $n$. Consider the space $V_k(M)$ of pairs $(x,\alpha)$ where $x \in M$ and $\alpha$ is a linear embedding $\mathbb{R}^k \hookrightarrow T_x M$ (or ...
1
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0answers
18 views

What is the most accessible reference on wall-crossing?

I am looking for a nice and easy to read reference on wall-crossing (in the context of Donaldson theory). Is there some accessible reference you have to suggest? I am interested in studying Donaldson ...
1
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0answers
53 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
3
votes
0answers
71 views

What is the formal definition of tangent hyperplanes?

My questions arise from 11.21 DEFINITON and 11.22 THEOREM ,both of them presented on p341 An Introduction To Analysis W.R.Wade 3ed. Question 1. Considering 11.21 DEFINITON, Let ...
1
vote
0answers
14 views

Non-degeneracy of curves/manifolds

Ok so I'm having some problems with understanding what it means for a manifold or curve to be non-degenerate. The definition I've been trying to get my head around is: "Non-degeneracy is a ...
3
votes
1answer
45 views

Cotangent bundle tensor product tangent bundle

What is the meaning of Cotangent bundle tensor product tangent bundle: $T^*M\otimes TM$? what will an element of this space be?
0
votes
1answer
77 views

degree of smooth maps from 2-sphere to 2-torus

Why any smooth map from the 2-sphere to the 2-torus has zero degree? Can we show that there is no surjective smooth map from 2-sphere to 2-torus?
0
votes
0answers
31 views

Submersion and some properties

Theorem: Let $f:M\to\mathbb R^m, M\subset\mathbb R^n$ be a submersion, $p\in M$ and $D_pf:\mathbb R^n\to\mathbb R^m$ is the functionalmatrix. Then there exists: - an open neighborhood $A$ on ...
3
votes
2answers
79 views

Is the pairing induced by the wedge product and integration nondegenerate on de Rham forms?

Let $M$ be a compact, oriented, smooth $n$-manifold and let $\Omega^*_{\mathrm{dR}}(M)$ be the commutative differential graded algebra of de Rham forms on $M$. We can define a pairing: \begin{align} ...
0
votes
0answers
25 views

Proving that the 'unit' basis vector fields for polar coordinates in the Euclidean plane are a noncoordinate basis.

This is Exercise 2.1 from Geometrical Methods of Mathematical Physics by Bernard Schutz. Show that the 'unit' basis vector fields for polar coordinates in the Euclidean plane, defined by A. $$ ...
0
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0answers
29 views

Comparison of orientations involving diagonals

This problem came up in a discussion about orientations and and seems more delicate than I expected: Let $M_1$, $M_2$ and $P$ be smooth oriented finite-dimensional manifolds without boundary. Let ...
1
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0answers
40 views

Show that $\partial(M\times N)=M\times\partial(N)$

Let M a $k$-dimensional manifold without boundary of $\mathbb{R}^{n}$ and N a $l$-dimensional manifold of $\mathbb{R}^{m}$ with or withour boundary. Show that $\partial(M\times N)=M\times\partial(N)$ ...
1
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0answers
66 views

Möbius strip parameterization and charts

A parameterization of the möbius-strip is given by : $$\begin{align}M=\{ (x,y,z) \in \mathbb R^3: x &= \cos t(1+ s\cos(t/2)),\\ y &= \sin t(1+ s\cos(t/2)),\\ z &= s\sin(t/2), \\ t ...
1
vote
0answers
48 views

Map from $\mathbb{R}^3 \rightarrow \mathbb{R}^6$ is Immersion for…

$$\phi :\mathbb R^3 \rightarrow \mathbb R^6$$ $$(u,v,w)\rightarrow \phi(u,v,w)=(x_1,x_2,x_3,x_4,x_5,x_6)$$ where $ \quad x_1=u^2 \quad x_2=v^2 \quad x_3=w^2 \quad x_4=vw \quad x_5=uw \quad x_2=uv$ ...
1
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0answers
34 views

convergence in the $C^r$-topology on $C^r(M,N)$ for $M$, $N$ compact manifolds

Let $M$ and $N$ be smooth (finite dimensional) manifolds without boundary. On the set $C^r(M,N)$ we choose the compact-open $C^r$-topology. This topology is defined as follows (I take the definition ...
7
votes
3answers
241 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
2
votes
1answer
128 views

Why do these two facts imply that $S^n$ is not parallelizable for $n$ even, $n \ge 2$?

Consider the following two facts. If $S^n$ admits a vector field which is nowhere zero, the identity map of $S^n$ is homotopic to the antipodal map. For $n$ even, the antipodal map of $S^n$ is ...
1
vote
0answers
34 views

Is this subset of $\mathbb{R}^{3}$ a topological manifold?

Consider the set $\mathcal{M}_{1} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ y = -1 \ \}$, this is a plane. Also consider the set $\mathcal{M}_{2} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ x=y=0 \ \}$, which ...
3
votes
2answers
112 views

$S^n$ admitting nowhere zero vector field implies identity map of $S^n$ is homotopic to antipodal map? [closed]

If $S^n$ admits a vector field which is nowhere zero, does it follow that the identity map of $S^n$ is homotopic to the antipodal map?
0
votes
1answer
44 views

How does the solution of a differential equation on a manifold yield a map?

In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind. In:"If the manifold ...
2
votes
1answer
47 views

Are the two standard descriptions of $\mathbb{C}P^{\infty}$ (topologically) equivalent?

While reading through some issues of Baez's (wonderful) "This Week's Finds in Mathematical Physics," I came across this statement (from week 149): $K(\mathbb{Z},2)$ is a bit more complicated: it's ...
1
vote
1answer
58 views

Prove S is a manifold.

At the moment the definition of a manifold I'm working with is that of a set $X$ equipped with a smooth atlas $A$. I want to prove that $\{(a,b)\in \mathbb{R}^n\times\mathbb{R}^n \mid a\cdot a=b \cdot ...
1
vote
1answer
21 views

Derivation of $f\in \mathcal C^1(M,N)$ where $M,N$ are smooth manifold.

I have a question about derivation of fonction $f:M\longrightarrow N$ where $M$ and $N$ are smooth manifold of dimension $n$. In my course, we try to compute $$\mathrm d_p f\left(\frac{\partial ...
0
votes
0answers
13 views

Show that the lemniscate is not a manifold. [duplicate]

Consider $\gamma(t)=(\sin(t),\sin(2t))$. How can I show that $\gamma(0)$ has no neighborhood homeomorphic to $\mathbb R$ ?
0
votes
1answer
30 views

A posteriori measures of numerical dissipation and dispersion

In PDEs, it is typical to find out how dissipative or dispersive a numerical method is by writing down the modified PDE corresponding to the numerical method, and seeing if that modified PDE contains ...