# 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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### last step in proof of existence of coordinate vector field

This is problem 7 on page 172 of Spivak's Differential Geometry pt. 1. Given a smooth manifold $M$ and a smooth vector field $X$ on $M$, Check that if the coordinate system $x$ is $x = \chi^{-1}$ ...
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### Is the smooth map with constant rank dense?

Let $M$ and $N$ be two Riemannian manifold. Assume that $f:M\rightarrow N$ is a smooth map and $\dim M < \dim N$. Can we find a smooth map $g$ $C^{k}$-close to $f$ which is immersive at each point ...
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### What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does "...
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### Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
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### Find the Poincare Dual of a ray in $\mathbb{R}^2-\{0\}$

This is example-exercise 5.16 in Bott and Tu (which I'm independently reading through.) The problem states: Let $M=\mathbb{R}^2-\{0\}$, and $X\subseteq M$ be the closed submanifold $\{(x,0):x>0\}$....
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### What is the definition of a properly-immersed arc?

I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have ...
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### Morse function on $\mathrm{SO}(n)$

I would like to prove that the following function is a Morse function, $F : \mathrm{SO}(n)\to\mathbb{R}$ $$A=(a_{ij})\mapsto\sum_{i=1}^na_{ii}\lambda_i$$ with $0<\lambda_1<...<\lambda_n$. ...
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### Euler characteristic of the closed unit ball

I would like to calculate the Euler-Poincaré characteristic of the closed unit ball $B$ by the de Rham cohomology, the Poincaré-Hopf theorem and Morse theory. de Rham cohomology: since $B$ is ...
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### Laplace-de Rham operator

Consider an operator $\partial = (-1)^k \star^{-1}\,d\star: \Omega^k(\mathbb{R}^n) \to \Omega^{k-1}(\mathbb{R}^n)$. Note that we equivalently can write $\partial = (-1)^{nk + n + 1} \star\,d\star$. ...
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### The Projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})$

So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a ...
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### Prove that $TTM$? is a orientable bundle on $TM$

I know that $TM$ is always orientable bundle. But what is $TTM$? How try to prove that $TTM$ is a orientable bundle on $TM$.