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

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3
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1answer
61 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
7
votes
1answer
82 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
3
votes
2answers
81 views

Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not ...
0
votes
0answers
14 views

spheres are not simpletic?

Reading some books on diferential geometry, a found that S^2n (with n>1) are not simpletic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not understand this ...
0
votes
1answer
25 views

Unsolved regular value problem in $\mathbb{R}^n$

I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold. My idea: Without loss of generality I ...
0
votes
1answer
19 views

Closure of $\{ ( e^t\cos t,e^t\sin t) : t \in \Bbb R \}$

Suppose $\alpha: \Bbb R\to \Bbb R^2$ given by $\alpha (t)=(e^t \cos t,e^t \sin t)$, $A=\alpha(t)$ is a smooth manifold. What is the closure of $A$? I know that the closure of the set is the set ...
1
vote
1answer
25 views

Prove that this application $f:S^n\rightarrow \mathbb{RP}^n$ is a local diffeomorphism, alternative approach using curves

I consider $f:S^n\rightarrow \mathbb{RP}^n$, the restriction to $n$-sphere $S^n$ of the canonical projection $\pi :\mathbb{R}^{n+1}\setminus \{0\} \rightarrow\mathbb{RP}^n $. I have to prove that $f$ ...
0
votes
0answers
48 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
1
vote
1answer
23 views

showing pushfarward

Let $M,N$ be two differentiable manifolds and $f:M \rightarrow N$ be a smooth map. Define a new map $F:M\rightarrow M\times N$ by $F(p)=(p,f(p))$ I can prove first part which is F is smooth but I can ...
1
vote
1answer
27 views

Given $f : P\rightarrow N$ $C^\infty$ and $\pi : M\rightarrow N$ local diffeomorphism show that $\tilde f$ s.t.$f= \tilde f \circ \pi$ is $C^\infty$

Let be $M$, $N$ and $P$ three differentiable manifolds. I consider $\pi: M \rightarrow N$ a local diffeomorphism and $f:P \rightarrow N$ differentiable. I have to prove that the application $$ ...
2
votes
1answer
24 views

Let $G=GL(n,\mathbb R)$, show that this application $ (A,B) \in G \times G \rightarrow AB \in G$ is $C^{\infty}$

Let be $G=GL(n,\mathbb R)$. I consider the application $$a: G \times G \rightarrow G$$ such that $$ (A,B) \rightarrow AB .$$ I have to prove that this application is $C^\infty$. I know the ...
1
vote
1answer
38 views

Induced Connection on $\Sigma\subset M$

Let $(M,g)$ be a Riemannian manifold, $\Sigma$ a manifold and $F:\Sigma \rightarrow M$ a smooth map. For $X,Y \in \Gamma(T\Sigma)$ vector fields and $\tilde{\nabla}$ the pull back connection on ...
0
votes
1answer
28 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
0
votes
1answer
70 views

What does it mean for a manifold to be oriented?

I'm currently working through Spivak's Calculus on Manifolds. I've got to Stokes' Theorem, which is stated thus (the bold is my emphasis): Stokes' Theorem If $M$ is a compact oriented ...
1
vote
0answers
16 views

Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
1
vote
1answer
37 views

How to show that a subset of $\mathbb{R}^2$ is a closed one-dimensional submanifold?

I'm trying to solve the following problem: For $c \in \mathbb{R} \setminus \{ 0 \}$, let $$C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2.$$ Show that for $c \neq 1/27$ the ...
3
votes
1answer
68 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
0
votes
0answers
30 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary ...
2
votes
1answer
40 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
1
vote
1answer
29 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
1
vote
1answer
33 views

Gluing together holomorphic functions on $\mathbb{P}^n$

The problem Let $U_j$ for $0\leq j\leq n$ denote the standard coordinate charts of the complex manifold $\mathbb{P}^n$. Fix $d\geq 1$ and assume we are given holomorphic functions $f_j:U_j\to ...
1
vote
0answers
35 views

Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...
4
votes
1answer
72 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
0
votes
0answers
20 views

Unbounded Geodesics and Nonpositive Curvature

I have the following interesting(?) question: Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics. As the question is ...
0
votes
0answers
24 views

Showing that the 2-torus is parallelizable

Here is the question Let $$ \widehat{\xi}: \mathbb{R}^2 \to \mathbb{R}^2 $$ be a smooth function satisfying $$ \widehat{\xi}(x,y)=\widehat{\xi}(x+m, y+n) $$ for all $x,y\in \mathbb{R}, ...
1
vote
1answer
33 views

Are PL-homeomorphic manifolds diffeomorphic?

Take two smooth manifolds. Since they are smooth, they both possess triangulations. Now assume that the triangulations are related by Pachner moves, that is, the triangulated manifolds are ...
1
vote
0answers
15 views

Non-affinely parametrized geodesics

Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent ...
5
votes
1answer
179 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 ...
1
vote
1answer
23 views

Fractional Sobolev spaces on closed manifolds

Let $M$ be a closed manifold and $0<s<1$. How is the fractional Sobolev space , $H^s(M)$ defined? In particular if $M$ is a closed smooth simple curve in the place how is $H^{1/2}(M)$ ...
1
vote
0answers
25 views

Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature ...
3
votes
1answer
79 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
1
vote
1answer
26 views

Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by ...
3
votes
1answer
76 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
0
votes
1answer
24 views

The continuity of a function defined by pieces

Let $M$ be a topological $n$-manifold and $\varphi \colon U\to \mathbb{R}^n$ a chart of $M$. Assume that $\varphi(U)=B(q, \varepsilon)$ is an open ball in $\mathbb{R}^n$ and that the map $f\colon B(q, ...
0
votes
2answers
33 views

CW -complex structure of boundary of a manifold

Given a CW-complex structure of manifold with boundary .Is there any natural way to construct CW-complex structure of its boundary? Thanking you.
1
vote
0answers
29 views

Invertible Matrices as a Manifold [duplicate]

I've been trying to prove that the set of all invertible $n \times n$ matrices is a differentiable manifold. My attempt is as follows: Define a map $\alpha : X \to XX^{-1} - I$ I take the inverse: ...
1
vote
0answers
42 views

Showing the Sum of $n-1$ Tori is a Double Cover of the Sum of $n$ Copies of $\mathbb{RP}^2$

I want to show that the non-orientable surface of genus $n$ has a 2-sheeted cover by an orientable surface of genus $n-1$. The base cases are easy: $S^2$ covers $\mathbb{RP}^2$ and I worked on a ...
2
votes
1answer
40 views

What does “orthogonal complement” mean in Milnor's Topology from the Differentiable Viewpoint?

Milnor writes on p. 11 If $M'$ is a manifold which is contained in $M$, it has already been noted that $TM'_x$ is a subspace of $TM_x$ for $x \in M'$. The orthogonal complement of $TM'_x$ in ...
7
votes
0answers
82 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 ...
0
votes
0answers
16 views

Adjoint representation for matrix groups (Gauge theory)

This is a question in regards to an identity in Gauge theory. Let $\omega$ be the connection one form on a principal bundle $\pi:P\rightarrow M$ and let $A_{\alpha}:=s_{\alpha}^*\omega$ be the gauge ...
0
votes
0answers
26 views

Sections of associated bundles isomorphism between spaces

I am reading some lecture notes which can be found here . They say that sections of $P\times_G F$ are represented by the functions $f:P\rightarrow F$ satisfying $f(pg)=\rho(g^{-1})\circ f$. Or ...
4
votes
1answer
29 views

Sections of associated bundles

Let $\pi:P\rightarrow M$ be a Principal bundle and $\pi_V:P\times_G F\rightarrow M$ be its associated bundle via the representation $\rho:G\rightarrow GL(V)$. Fact: $\Gamma(P\times_G ...
1
vote
1answer
39 views

Winding number of differential curve

Consider the one-form $\omega$ on $\textbf{R}^2$\ {(0,0)} defined by $\omega$ = $\frac{xdy-ydx}{x^2+y^2}$ Let K $\subset$$\textbf{R}^2$\ {(0,0)} denote the positive x-axis. Let $\gamma$ : $[a,b]$ ...
1
vote
0answers
50 views

Gradient in local coordinates on a manifold with Riemannian metric

Let $M$ be a smooth manifold with a Riemannian metric g : $TM\otimes TM$ -> R If f is a smooth function from M to R, the gradient of f with respect to g is the vector field $\nabla f$ defined by ...
0
votes
0answers
28 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
2
votes
2answers
45 views

Exterior differentiation of one form on a smooth manifold

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I'm fine with the right side of the equation, ...
1
vote
1answer
49 views

Guillemin & Pollack's proof on Whitney embedding theorem

I am confused with a little detail in Guillemin & Pollack's proof on Whitney embedding theorem. Please see page 54 in their book "Differential topology". In the second paragraph of page 54, they ...
0
votes
1answer
25 views

Quotient of a cylinder by the product of a rotation and translation

Let $D$ be the unit disk, and $$f : D \times \mathbb R \rightarrow D \times \mathbb R$$ be defined by $$f(z,t)=(e^{2i\pi \alpha}z, t+1)$$ where $\alpha$ is an irrational number. Now consider $X = (D ...
0
votes
1answer
33 views

Finding coordinate charts from a torus (square with sides identified) to $\mathbb R^2$.

I have a torus which is thought of as the unit square with sides identified in the usual manner. I know that for every point on the square I have to find a neighborhood and a diffeomorphism to open ...
0
votes
0answers
20 views

A complicated question on barycenter map

I have $\Omega\subset \mathbb{R}^n$ is an open bounded subset, i define for $r>0$ the set $\Omega_{r}^{+}=\{x\in \mathbb{R}^n, d(x,\Omega)\leq r\}$ we suppose that $0\in \Omega.$ I have also this ...