Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [differential-topology]

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.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

7287 questions
0
votes
1 answer

Bases $\{v_1,…,v_k \}$ and $\{ v_1' ,…,v_k' \}$ for $V$ are equivalent iff $T(v_1,…,v_k )$ and $T(v_1',…,v_k' )$ have the same sign

a) Let $T$ be a non zero element of $∧^k (V^*)$ where $\dim⁡ V=k$. Prove that 2 ordered bases $\{v_1,…,v_k \}$ and $\{ v_1' ,…,v_k' \}$ for $V$ are equivalent oriented if and only if $T(v_1,…,v_k )$ and $T(v_1',…,v_k' )$ have the same sign. b) …
Diane Vanderwaif
  • 4,292
0
votes
1 answer

Subsets of a manifold.

I read that every open subset $A$ of a manifold $M$ is a submanifold (it is a manifold with the induced topology by $M$). If I understand correctly, the argument is that, for an element $x \in A$, one can considerer a chart $(U,f)$ in $M$, and…
ted
  • 3
0
votes
2 answers

Smooth maps homotopic to the inclusion - updated.

First problem - (my original question before the editing) Prove or disprove the following: Let $A$, $B$ be differentiable manifolds such that $A \subseteq B$, and $s: A \to B$ a smooth map. Then $s \sim i$ where $i: A \to B$ is the inclusion…
Biagio
  • 393
0
votes
1 answer

Prove that a map $f:S^1→S^1$ extends to the whole ball $B=\{|z|≤1\}$ if and only if $deg(f)=0$

Prove that a map $f:S^1→S^1$ extends to the whole ball $B=\{|z|≤1\}$ if and only if $deg(f)=0$ again I can only do one direction => $f:S^1→S^1$ is smooth, and $S^1 = \partial B$. Assume that $f:S^1→S^1$ extends to the whole ball$B$ then $deg(f)=0$…
Diane Vanderwaif
  • 4,292
0
votes
0 answers

Orientability of integrable plane fields

Let $\xi \subset \text{T}M$ be a integrable plane field on a smooth 3-manifold (i.e. the tangent field of a foliation). Is it true that $\mathcal{F}$ is coorientable?
Antonio Alfieri
  • 1,555
  • 7
  • 13
0
votes
1 answer

Tubular neighborhood of $X^k$ compact submanifold with normal bundle $\perp X$ trivial

For $X^k\subset M^n$ compact submanifold with $\perp X$ trivial and set $S^k$ the $k$-sphere. Then there is a function $f:M^n\rightarrow S^k$ such that $X$ is the preimage for a regular value. My question is: the converse is true?
Donyarley
  • 818
0
votes
1 answer

show that $R^n -X$ has at most 2 connected components.

show that $R^n -X$ has at most 2 connected components. Theprem: Suppose that $X$ is the boundary of $D$, a compact manifold with boundary and let $F:D \to R^n$ be a smooth map extending $f$; supposed that $z$ is a linear regular value of $F$ that…
Diane Vanderwaif
  • 4,292
-1
votes
1 answer

Differential forms on tensors

With T: R^m -->R^n be linear transformation T(x) = B*x and if psi sub I is an elementary alternating k-tensor on R^n, then T*psisub I has the form: $$ T^**\psi_I $$ = sigma sub [J] cJ*psi[J] where psiJ are the elementary alternating k-tensors on…
mary
  • 2,374
1 2 3
…
40
41