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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.

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Looking for two non diffeotopic embeddings $\mathbb{R}\rightarrow\mathbb{R}$

im studyin the book "Introduction to differential topology" by Th. Bröcker and K. Jänich and I'm stuck with two of the exercices in Chapter 9 on isotopy. A short overview (for those who are familiar with the topic: skip until the fat word): Let…
Takirion
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Parallel transport perspective of gauge transformation invariance for connections

Defining a connection on a principal $G$-bundle $P \to M$ is equivalent to defining a parallel transport on $P$ along curves in $M$. With this perspective, Ralph Cohen commented in his notes on the topology of fiber bundles (pp.62) that a…
PhysicsMath
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partial differentiation on a manifold

im currently taking up a course on differential geometry and the last topic is topology. id like to ask for help in our homework since im kind of new to this kind of questions which involves proving since im from the physics department and im taking…
anonymous
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Exterior derivative as a special case of covariant derivative?

In terms of local coordinates we can make a covariant derivative exterior-derivative-like (this is actually Levi-Civita connection) \begin{equation} \Gamma_{i,j}^k = \Gamma_{j,i}^k \implies D(dx_k) = \sum_{ij}{\Gamma_{i,j}^k dx_i \otimes dx_j} = 0…
PhysicsMath
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Does a map with fibers $S^2\vee S^2$ have to be a locally trivial $S^2\vee S^2$ bundle?

Let $X\to Y$ be a proper map between pseudo-manifolds such that fibers are $S^2\vee S^2$, is it true that $X\to Y$ is locally trivial $S^2\vee S^2$ bundle?
user93417
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A trivial vector bundle with a riemannian metric has an isomorphism with the trivial bundle that is an isometry on each fiber.

The problem: Demonstrate that If $\xi^n=(E,\Pi,M)$ is a trival vector bundle with a riemannian metric, then there exist an isomorphism $\psi:E\to$ M x $R^n$ Such that $\psi $ is an isometry on each fiber. What I have got: Since $\xi^n$ is a trivial…
Alicia Basilio
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Is infinity always a regular value of rational function from $S^1\to S^1$

$f$ is a rational function $S^1\to S^1$, where $S^1$ is the one point compactification of $\mathbb R$. Is $\infty$ always a regular value of $f$?
user136592
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Derivations having local character

Given a (smooth) manifold, it is known that derivations $D:C^\infty(U) \to C^\infty(U)$ on a chart $(U,\kappa)$ are equivalent to a vectorfield on $U$, i.e. to an element $X \in \Gamma(TU \to U)$. The definition of a derivation is that $D$ is…
user42761
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Exhibit a smooth map $f : \mathbb{R} \to \mathbb{R}$ whose set of critical values is dense.

Question 1.7.5 (Differential Topology - Guillemin and Pollack) Exhibit a smooth map $f : \mathbb{R} \to \mathbb{R}$ whose set of critical values is dense. From [Exercise 1.1.18], there is a function $g : \mathbb{R} \to \mathbb{R}$ such that $g(x) =…
user230283
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Lemma 2 - p.204 (Differential topology - Guillemin and Pollack)

Let $A$ be a compact subset of $\mathbb{R^n}$. Suppose that $A \cap V_c$ is contained in an open set $U$ in $V_c$. Then for any suitably small interval $I$ about $c$ in $\mathbb{R}$, $A \cap V_I$ is contained in $I \times U$. $V_c$ is the…
user230283
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$f$ integrable vs $\int_Af$ exist - in Spivak's Calculus on Manifolds

I am confused by the difference between "$f$ integrable" and "$\int_Af$ exist", in Spivak's notion of extended integral. Here's his definition. Note that it has a flaw: each $\varphi $, in the partition of unity $\Phi$ subordinate to $O$, should be…
JSCB
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Orientability of Orbit Space of a Group of Diffeomorphism

While working on some problem in differential topology, I had to prove the lemma below. It seems to me like I have not needed all the requirements in the lemma, leading me to think that I have missed some of the finer points. Question Given a…
JP-Ellis
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Problem about perfectly convex set and convex set.

Let $W$ be a subspace of a Banach space $X$. Which of the following are true. a. W is closed then it is perfectly convex b. W is perfectly convex then it is closed. Definition of perfectly convex set says that, a subset A of X is called perfectly…
S. Pitchai Murugan
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Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is. - "Differential topology" of Guillemin and Pollack

I know that we can refered to the question How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?. I have the same question with an answer, and I needed that someone tell me if it is good or not (why?) Here's the question : The…
user230283
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Is the $C^0$-fine topology finer than the metric topology?

Let $C(E,F)$ be the set of continouos maps between metric spaces $E$ and $F$. Suppose we are given the $C^0$ fine topology and a metric topology on $C(E,F)$. We know that the fine topology is finer than the compact-open topology, but is it finer…
mathphy
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