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.
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Integration of differential $k$-form along continuous singular $k$-chain
Suppose that $M$ is a smooth manifold, and let $\omega$ be a fixed differential $k$-form.
Let $C$ be a singular $k$-chain.
My goal is to define $\int_C \omega$, which is usually impossible because $C$ ...
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Extending a local embedding to a neighborhood of a submanifold
Let $S \subset M$ be a submanifold, and let $f : M \to N$ be a smooth map such that:
$f|_S : S \to N$ is an embedding;
for every $p \in S$, there exists an open neighborhood $U(p) \subset M$ such ...
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Reference request: Pseudogroup of homeomorphisms give class of manifolds
I'm looking for a reference that shows how a pseudogroup of homeomorphisms can produce a class of manifolds. In particular, I'm interested if this data provides a notion of immersions/submersions in ...
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Existence of smooth bump function without using partition of unity
This problem is an exercise of my course.
My professor does not put second-countability on the definition of smooth manifold.(Our "smooth manifold" is a locally Euclidean Hausdorff space ...
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Is this gradient defined? (gradient of a probability density function (pdf) where the pdf is w.r.t a non-Lebesgue measure)
Let $\nu$ be a measure on the unit-circle that is absolutely continuous with respect to the spherical measure $\sigma^1$.
Let $p$ be the density function admitted by $\nu$, so that:
$$\nu(A)=\int_{S^1}...
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Representatives for Components of Kunneth Formula in de Rham Cohomology
I am working with a differential form $\omega$ over product manifold $X\times Z$, $X$ and $Z$ both compact connected manifolds, where I am able to show the following:
Let $\{x_1,\dots,x_k,z_1,\dots,...
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Reconciling some obstruction-theoretic definitions of the first Chern class in Gompf and Stipsicz
I am currently working through Chapter 1.4 of Gompf and Stipsicz, and I am confused about their definition(s?) of the first Chern class via obstruction theory.
My understanding of this definition (...
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Why is it so difficult to answer such a basic question in this field? ("Basic Manifold Theory" by Yukio Matsumoto.)
I am reading "Basic Manifold Theory" by Yukio Matsumoto:
Even if a differential structure exists on a topological manifold, it is not necessarily unique. The first discovery of an example ...
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What precisely is the weak Whitney $C^k$ topology?
In Hirsch, Differential Topology, chapter 2, section 1, he defines the weak Whitney $C^k$ topology on $C^k[M, N]$ where $M, N$ are $C^k$ manifolds by giving a sub-basis. I'm unclear on one point on ...
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Critical points of perturbed Morse function
I'm currently working on Exercise 1.7.19 in Guillemin and Pollack's Differential Topology and have reached something of an obstacle. The exercise asks to show that if $X$ is a compact submanifold of $\...
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Frobenius theorem in Ralph Abraham's Foundations of Mechanics
I am studying the proof of Frobenius theorem in Ralph Abraham's Foundations of Mechanics. The theorem states that a subbundle $E \subset TM$ is integrable if and only if it arises from a regular ...
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Proving a property holds on a meager set
Let $\Theta \subset \mathbb{R}^2$ be an open, bounded set with a representative element $(\alpha,\beta)$.
Let $\mathcal{X} = C^2(\Theta)$ be a function space endowed with the Whitney $C^2$ topology. ...
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Which notions and theorems of singularities of a vector field are used in this delightful comic?
This question is, I think, about vector fields and their relation to the Euler (-Poincare-Borel-Moore) characteristic, but I don't think what I'm looking at is immediately understandable via the ...
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Constant-length logarithms of diffeomorphisms at a fixed time step [closed]
Let $(M,g)$ be a closed smooth Riemannian manifold and fix $\delta > 0$. Given a smooth diffeomorphism $f: M \to M$:
Question: Does there exist a smooth vector field $V$ on $M$ with constant ...
DimensionalBeing
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Can "simply connected" be characterised by $C^1$-contractibility of $C^1$ simple loops?
(This question is loosely similar to, but clearly distinct from, Bounded and open simply connected region is smoothly contractible.)
Let $S=\{x \in \mathbb{R}^2 : |x|=1\}$ and $V=\{x \in \mathbb{R}^2 :...
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