Questions tagged [manifolds]

Question 1

I am trying to get a better grasp of how to find the basis of the tangent space. Here is one example I worked on in hopes of practicing it: Consider the chart $(U,\psi)$, the manifold $\mathcal{M} = S^...

Question 2

Let $M = \{(x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb{R}^{4}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2} = -1\}$. Prove that $M$ is a regular submanifold of $\mathbb{R}^{4}$ with dimension 3. Compute the ...

Question 3

I am writing because in our first course on differential geometry in Spain, our professor has decided to use the concept of triangulation to define integrals of n-forms on differential manifolds, but ...

Question 4

The motivation to my question is the conflicting definitions of a covering spaces presented in this question. The answer to the linked question is the following; Angina Seng's Answer; Hatcher and ...

Question 5

As I understand it, a color model (like the RGB model) is a way of mapping the space of all human-perceptible colors to a certain manifold -- typically, but not always, either three- or four-...

Question 6

I am trying to understand the proof that the homeomorphism problem of compact 4-manifolds is undecidable. The proof relies upon on two facts. Firstly, that one can construct a finite 2-dimensional CW ...

Question 7

I’m trying to understand solvability of $$ L_X u = f $$ for a vector field $X$ on a manifold, and then compare this with some related linear problems for left-invariant forms on a Lie group (where ...

Question 8

I wish to describe a problem encountered in my research and am seeking advice, or just pointers on where to look. My setting is as follows. Given a scatterplot of data in $\mathbb{R}^D$, we wish to ...

Question 9

Let M be an n-dimensional manifold, and let $(\pi_\alpha: U_\alpha \to V_\alpha)$ be an atlas of coordinate charts for M, where $U_\alpha$ is an open cover of M and $V_\alpha$ are open subsets of ${\...

Question 10

Let $M$ be a closed oriented smooth manifold of dimension $d$, and let $G$ be an abelian group. Then the cohomology group $H^n(M;G)$ corresponds bijective to set of homotopy classes of based maps $[M,...

Question 11

This is problem 20.1 from Loring Tu's An Introduction to Manifolds: Let $I$ be an open interval, $M$ a manifold, and {$X_t$} a 1-parameter family of vector fields on $M$ defined for all $t \neq t_0 \...

Question 12

Say $E = \{p_\theta : p_\theta(x) = \exp(x^\top \theta - A(\theta)), \theta \in \Theta, M \theta = b\}$ is an exponential family affinely constrained in its natural parameter, where $\Theta$ is a ...

Question 13

I am currently studying Lie groups and manifolds using these notes. As a follow up to this question (in which I asked how the partial derivative of a regular function is defined with respect to local ...

Question 14

Recently I came across the definition of paracompactness (while reading about manifolds). Here are the relevant definitions for paracompactness: Let $X$ be a topological space. Definition (cover, open ...

Question 15

Let $M$ be a smooth connected manifold of dimension $\geq 2$, and let $\phi: \mathbb{R} \times M \to M$ be a complete flow on $M$. Suppose there exists a point $x_0 \in M$ whose orbit is dense in $M$, ...

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Questions tagged [manifolds]

Confusion regarding Tangent Basis

Connected components on a regular manifold

References for triangulation of manifolds [closed]

Connectedness and the equivalence of covering-space definitions

What mathematical structure do the manifolds described by color models and color spaces have? [closed]

Construction of family of manifolds in showing that homeomorphism problem of compact 4-manifolds is undecidable

Invertibility of Lie Derivative

geodesic distance in embedded manifolds

Borel-$\sigma$ algebra on a manifold

Cohomology classes on manifolds induce homology and bordism classes of Eilenberg-MacLane spaces

The limit of a family of vector fields is independent of chart

Does online mirror descent between dually flat space converge to the global optimum

The directional derivative in local coordinates - definition verification

Understanding the definition of paracompactness

Does a dense orbit imply topological transitivity for flows on manifolds?

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