# 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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### The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
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### Finding bump function on a smooth manifold using partitions of unity.

Let $M$ be a smooth manifold. Let $A$ and $B$ be disjoint closed sets of $M$. Show there exists a smooth function $f$ such that $f^{-1}(0)=A$ and $f^{-1}(1)=B$. This is my idea so far, Since $A$ ...
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### Exponential map on Diffeomorphism group of $S^1$

I am reading Segal book on Loop groups, and he mentions the following theorem: $$\exp: Vect(S ^1) \rightarrow Diff(S ^1)$$ the map taking a vector field to the diffeomorphism obtained by flowing ...
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### A question about contractibility of a Lie group

In Lawson's book Spin Geometry, chapter II, before Proposition 1.11, it mentions a fact that if the 1st, 2nd and 3rd homotopy groups of a Lie group $G$ vanish (although $\pi_2(G)=0$ automatically), ...
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### Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$

Supposed that the derivative of $f:X\to Y$ is an isomorphism whenever $x$ lies in the sub-manifold $Z \subset X$, and assume that $f$ maps $Z$ diffeomorphically onto a $f(Z)$ . Prove that $f$ map a ...
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### De Rham Cohomology is Sheaf Cohomology

I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold. I would like to do this without using Čech ...
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### Noncomplete riemannian manifolds

In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to ...
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### Divergence in terms of Levi-Civita connection

The divergence of a vector field $X$ on a manifold $M$ is defined usually as the function $\text{Div}(.)$ such that $(\text{Div} X) \;\mu =L_X \mu$ for $\mu$ a volume form. I know that there is also ...
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### Prob. 10 of Chapter 8 of Milnor's Topology From a Differentiable Viewpoint. (Tangent Bundle is a Smooth Manifold.)

I want to solve Prob. 10 of Chapter 8 from Milnor's Topology From a Differentiable Viewpoint. The problem states that: Let $M\subseteq \mathbf R^k$ be a smooth manifold of dimension $m$. Show ...
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### Does the cup product on de Rham cohomology induce a nondegenerate bilinear form?

I have small issue I came across in the following. Suppose $M$ is a compact, oriented manifold of dimension $4n+2$. I want to prove that the de Rham cohomology group $H^{2n+1}(M)$ are even ...
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### Questions about simplex and affine space

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
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### Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
I'm trying to study the book "Manifolds of Differentiable Mappings" written by Michor and I came across the following: He considers two smooth manifolds $M$ and $N$ and define an equivalence relation ...