# 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.

280 views

### Uniqueness of integral curves.

Suppose we have a smooth compact manifold $M$ with boundary and a vector field $X$ on $M$. The maximal integral curves on $M$ are unique and hence their images give a partition of $M$. Let $p\in M$ ...
106 views

75 views

### $d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$

If $f: S_1\to S_2$ is a smooth map betweentwo regular surface $S_1$ and $S_2$, then $d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$ I cannot prove this statement. I think that ...
97 views

### Transversality and intersection mod.2

I am troubled by the following fact: If $f,g : X\to Y$ are homotopic and both transversal to $Z$ then the mod.$2$ intersection numbers are equal $I_2(f,Z)=I_2(g(Z)$. (the book by Guillemin and ...
58 views

### What is the type of $u$ in this definition of knot?

I am going to quote from the second paragraph of the Introduction of Möbius Energy of Knots and Unknots by Michael H. Freedman, Zheng-Xu He and Zhenghan Wang. Let $\gamma = \gamma (u)$ be a ...
254 views

### Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
43 views

### Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...
238 views

### Smooth diffeomorphisms preserving common symmetries

Let $A$ and $B$ be two $C^\infty$ orientation-preserving diffeomorphic, connected, bounded, open subsets of ${\mathbb R}^n$, with finitely many ends and $G$ the group of isometries of ${\mathbb R}^n$ ...
293 views

### Prove the directional derivative operators at a point on manifold form a vector space

One of the way to define tangent space is to use directional derivative. However, it's not clear at the first glance that the directional derivative operators form a vector space. Let $D$ be the set ...
95 views

### Constructing a non-degenerate vector-field from old one

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of the ...
172 views

### Existence of differential form on a manifold

I have a fundamental question about the existence of differential forms on manifolds. A $k$-form on a manifold in local coordinates looks like $f(x_1,...,x_n)dx_{i_1}...dx_{i_k}$, where $f$ is a ...
46 views

### Why is $\frac{f}{\|f\|}$ a submersion when this matrix has rank $k$?

A paper I'm reading defines the following $\left(\frac{(k-1)k}{2}+1\right)\times n$ matrix: \begin{align*} \Omega_f(x) := \begin{bmatrix} \omega_{1,2}(x)\\ \vdots \\ \omega_{i,j}(x) \\ \vdots \\ \...
245 views

### Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
127 views

### Intersection of lines with compact smooth manifolds

everybody. I need a hint on this: I have to prove that a compact smooth submanifold of R^n intersects almost every one dimensional linear subspace (in R^n) in a finite set of points. I know I have to ...
47 views

### The space of regular curves deformation retracts onto the space of arclength parameterized curves

Let X denote the space of smooth maps from the circle into R^3 which have no zero derivatives. This is an open submanifold of the Frechet space of smooth maps from the circle into R^3. Let Y denote ...