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Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Vector fields- differential topology

Can anybody please explain me the reason for last 6th line. (The set of all vectors at all points...)
User
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Let $f : M\to N$ be a submersion with $M$ compact and $N$ connected. The $f$ is surjective.

I have no idea how to do this. I tried think in, once $N$ is connected and locally path connected it has to be path connected, but, this does not help. Any hints, solutions, will be very appreciate... Now I realized that $f(M)$ will be also…
L.F. Cavenaghi
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Being morse function for a determinant map on M(n)

Show that the determinant map on M(n) is Morse function if n=2. I know that f to be a morse function, all critical points for f must be nondegenerate. But i dont know how i calculate the derivative of a determinant map
diffgeo
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Prove that $x\in\text{Bd}(U)\iff f(x)\in\text{Bd}(V)$

$\newcommand{\Fr}{\text{Bd}}$ I denote $\text{Bd}(A)$ the boundary of the et $A$. Let $U,V\subset \mathbb R^n$ open and $f:\overline{U}\longrightarrow \overline{V}$ an homeomorphism. Therefore $$x\in\text{Bd}(\overline{U})\iff f(x)\in…
idm
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Question about notation with differential one-form

Let $M$ be a manifold. The differential of $f \in C^\infty(M)$ is defined as the one-form $df$ such that $(df)(v) = v(f)$ for every tangent vector $v$ to $M$. It is then stated that if $f \in C^\infty(M)$ and $h \in C^\infty(\mathbb{R})$, then…
Randy Randerson
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Tangent spaces and derivatives are defined in the setting of manifolds with boundary

First, suppose that $g$ is a smooth map of an open set $U$ of $\mathbb{H}^k$ into $\mathbb{R}^l$. If $u \in \partial U$, the smoothness of $g$ means that it may be extended to a smooth map $\phi$ defined in an open neighborhood of $u$ in…
user230283
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Show that the natural copy of $\mathbb{R^{n-1}}$ inside $\mathbb{R^{n}}$ - namely, $\{(x_1, x_2,..., x_{n-1},0)\}$ - has measure zero

Question P.202 (Differential Topology - Guillemin, Pollack) : Show that the natural copy of $\mathbb{R^{n-1}}$ inside $\mathbb{R^{n}}$ - namely, $\{(x_1, x_2,..., x_{n-1},0)\}$ - has measure zero. [Hint : Show that every compact subset of…
user230283
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Let $U$ be an open set of $R^n$, and let $f: U \to R^n$ be a smooth map. If $A \subset U$ is a measure zero, then $f(A)$ is of measure zero.

Let $U$ be an open set of $R^n$, and let $f: U \to R^n$ be a smooth map. If $A \subset U$ is a measure zero, then $f(A)$ is of measure zero. Proof (Differential Topology - Guillemin and Pollack): $ \textbf{We may assume that}$ $\overline{A}$ $…
user230283
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What is the meaning of $\Omega^o_n$?

First, write $M^n \sim M^n$ cobordant, if $M^n$ # $M^n = \partial W^{n+1}$. Where # represent connected sum. Then define $ \Omega^o_n = \{ \textrm{closed manifolds} \} / \sim$ From $M^n$ # $S^n$ = $M^n$, This gives \begin{align} \Omega^o_1 = 0,…
phy_math
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Subset of a smooth manifold

I am actually in the resolution of the problem Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is. - "Differential topology" of Guillemin and Pollack (my own question), and I was wondering if a subset of a smooth manifold…
user230283
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Induced bundles and Smooth Maps

Given a smooth map between compact manifolds without boundary what criteria guarantee that the induced bundle is isomorphic to the tangent bundle? And less generally, suppose the map has non-zero Brouwer degree. What then?
Joe S
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Prove each coordinate is a differentiable function

This is an exercise from a book called "Differential Topology" 2-11: Let $M$ be the sphere $x^2+y^2+z^2=1$ in 3-space. Prove that each of the Euclidean coordinates $x,y,z$ is a differentiable function on $M$. I know how to show a function is…
SamC
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Expressing derivative as linear combination of derivatives of coordinate functions?

This is an old exam problem at my school: Let $F\colon M\to\mathbb{R}^k$ be a smooth map of smooth manifolds, with coordinate functions $F^1,\dots,F^k$. Let $c\in\mathbb{R}^k$ be a regular value of $F$, and $C=F^{-1}(c)$. If $f\colon…
YN Chew
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Applying Lie bracket to a smooth function.

Let $\textbf{X}$ and $\textbf{Y}$ be vector fields on an $n$-dimensional manifold, $M$. Let $f^{-1} : M \to \mathbb{R}^n$. We can represent $$\textbf{X} = a^1 \frac{\partial}{\partial x^1} + \dots + a^n \frac{\partial}{\partial x^n}$$ and $$…
Yuugi
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Show that the volume element of $V$ is $ϕ_1\wedge\cdots\wedge ϕ_k$.

a) Let $V$ be an oriented $k$-dimensional vector subspace of $\mathbb{R}^N.\,$Prove that there is an alternating $k$-tensor $T\in\bigwedge^k (V^*)$ such that $T(v_1,\ldots,v_k )=1/k!$ for all positive oriented ordered orthonormal base.…
Diane Vanderwaif
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