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

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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 ...
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1answer
14 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 $$ ...
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1answer
20 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 ...
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1answer
18 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 ...
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42 views

A compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary

Under what conditions is it true that a compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary (or more generally, when a manifold is embedded in some topological space)? For ...
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1answer
19 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 $\partial \vec r *A$ ...
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38 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 ...
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12 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 ...
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1answer
30 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 ...
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1answer
39 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 ...
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28 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 ...
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1answer
36 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.
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36 views

What is the slow manifolds? and how to calculate?

I'm a newbie in slow manifolds and dynamical system. I cannot understand the concept of slow manifolds and how to calculate that. Please explain the concept of slow manifolds intuitively and ...
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1answer
27 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 ...
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0answers
22 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 ...
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31 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 ...
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0answers
42 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 ...
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0answers
17 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 ...
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0answers
19 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}, ...
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1answer
26 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 ...
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11 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 ...
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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 ...
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1answer
19 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 $H^{1/2}(M)$ ...
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21 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 ...
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24 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 $C^1$ ...
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1answer
24 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 ...
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1answer
63 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 ...
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1answer
22 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, ...
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1answer
22 views

Prove $M$ is a manifold [closed]

Let $ M$ be the set of all points $(x, y, z) \in \mathbb{R}^3$ satisfying both of the equations $x^2 + y^2 = z^2 + 1$ and $x + y + z = 0$. Prove that $M$ is a manifold. What is the dimension of $M$?
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30 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.
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28 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: ...
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34 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 ...
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1answer
34 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 ...
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65 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 ...
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12 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 ...
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22 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 ...
3
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1answer
22 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 ...
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1answer
31 views

Tangent vectors to a space of parametrization maps and vector fields

In studying problems of locomotion of deformable bodies in highly viscous fluids I found something that although I have an intuition about I don't know how to make it mathematically rigorous. The ...
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1answer
37 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 $\gamma$ : $[a,b]$ ...
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41 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 ...
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0answers
27 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 ...
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2answers
44 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, ...
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1answer
44 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 ...
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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 ...
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1answer
28 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 ...
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18 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 ...
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1answer
50 views

Passage in a proof of a lemma

Here is a lemma and a proof given to me in class. Lemma If $M$ is a smooth manifold, $K\subseteq M$ a compact subset, $A\subset M$ an open set containing $K$< then there exists a compact-support ...
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2answers
60 views

$\Delta_L(\text{im}\,\delta^*_g)\subset\text{im}\,\delta^*_g$ and $\Delta_L\big(\text{ker}\,\text{Bian}(g)\big)\subset\text{ker}\,\text{Bian}(g)$?

Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The Lichnerowicz Laplacian of $g$ is the map ...
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0answers
52 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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2answers
110 views

Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...