# Tagged Questions

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.

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### What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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### Derivative of vector field on a manifold sends the tangent plane to itself

A vector field $\vec{v}$ on a (smooth) manifold $X \subset \mathbb{R}^N$ is a map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)\in T_x(X)$ for every $x \in X$. Suppose $\vec{v}(x)=0$, show that ...
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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... ...
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### Example of a disconnected manifold where the tangent space is not the dimension of the manifold?

Wikipedia says that the tangent spaces of a connected manifold all have the same dimension, equal to that of the manifold. Well, is there an example of a simple disconnected manifold that doesn't ...
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### De Rham cohomology ring of flag bundles/manifolds in Bott and Tu

I'm trying to understand the result for the de Rham cohomology ring of flag manifolds in Differential Forms in Algebraic Topology by Bott and Tu. I'm sort of starting from the result and working ...
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### $f: M \to Y$ $C^{1}$ map between manifolds $M$ e $N$ with dimension $m$ and $n$ respectively then $f$ is locally proper

We say that $f:f: M \to N$ is locally proper if for all $x \in M$ there exist $V \ni x$ open in $M$ such that $f|_{\overline{V}}:\overline{V} \to Y$ is proper. I know two ways to prove it. First, ...
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### Bundle that is isomorphic to the bundle of Whitney sum

I am involved with one question that a friend of mine asked. If a vector bundle $(E,\pi,M)$ has two sub-bundles, $(F,\xi,M)$, $(L,\psi,M)$, and $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ ...
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### Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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### compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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### Normal bundle of sub-manifold is a manifold

This is exercise 2.3.12 of Guillemin and Pollack: Let $Z$ be a sub-manifold of $Y$, where $Y \subset \mathbb{R}^M$. Define $N(Z;Y)=\{(z,v):z\in Z, v\in T_z(Y), v \perp T_z(Z)\}$. Prove that $N(Z;Y)$ ...
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### Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
Let $X$ be a manifold lying in $\mathbb{R^n}$, with $dim(X) = n-1$. Define $h: N(X) \to \mathbb{R^n}$ by $h(x, v) = x + v$, where $$N(X) = \{(x, v) \in X \times \mathbb{R^n}: v \perp T_x(X)\}$$ is the ...