# 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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### $f: M \to Y$ $C^{1}$ 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 ...
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### Do the level sets of a constant rank map give a foliation of the domain?

I'm working through Smooth Manifolds by Lee and came to problem 19-8, where you're asked to prove that the level sets of a submersion form a foliation of the domain. However when I try to solve this ...
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### A new definition of a focal point

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 ...
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### Immersions are open maps

Let $M,N$ be manifolds with $\dim M = \dim N$. If $f:M\to N$ is an immersion then $f$ is open. I thought that I have solved it, but then I thought there could be a mistake: Let $p\in M$. As $f$ is ...
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### Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
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### Embedding of $2$-torus minus one point into $\mathbb{R}^2$ as an open set?

Let $M^2$ be the $2$-torus minus one point. Is there an embedding of $M^2$ into $\mathbb{R}^2$ as an open set?
### Can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$? [duplicate]
As the question title suggests, can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$, i.e. in codimension $1$?
### Immersion of $M^n$ into $\mathbb{R}^n$, is $M^n$ orientable? Compact? [closed]
Say we have an immersion of $M^n$ into $\mathbb{R}^n$ (same dimension). I have two questions. Is $M^n$ orientable? Is $M^n$ compact? Thanks in advance!