# Questions tagged [manifolds]

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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### Random Walk on a Discrete Circle of n-dimensions and a Differentiable Manifold of n-dimensions

Let us have a circle of k discrete points. We will walk on this 2 dimensional circle randomly with a 50%-50% distribution to go clockwise or counterclockwise. If we take the number of steps we take to ...
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### Is a monomorphism of vector bundles an embedding?

Let $(p,E,B)$ and $(p',E',B)$ be two $C^r$ vector bundles over the same base space $B$. (When $r>0$ all the spaces are $C^r$ manifolds and all the maps are $C^r$ smooth. When $r=0$ they are just ...
1 vote
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### An example of a non hausdorff space

Here is the question I am trying to understand its solution: Show that there exist nonorientable $1$-dimensional manifold if the Hausdorff condition is dropped from the definition of a manifold. I ...
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### Basic questions about the flat torus

I am trying to undesrtand a bit better the geometry of the flat torus $T^n=\mathbb{R}^n/{\mathbb{Z}^n}$ and during this process some question that I would like to solve have arising: Regarding the ...
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### What is the difference between a parameterization of a manifold and a local chart?

I am confused on the difference between a parameterization of a manifold $M$ and local charts on $M$. If $M$ has dimension $n$, we may find a subset $U \subset \mathbb{R}^n$ such that there exists a ...
1 vote
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### Closed curve in a torus diffeomorphic to a circle?

In Example 15.9 of Tu's "An Introduction to Manifolds: Second Edition", it is written Let G be the torus $R^2/Z^2$ and L a line through the origin in R2. The torus can also be represented ...
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### Differential forms, antisymmetry, and integrals

Is it appropriate to think of surface integrals and volume integrals that come up in physics as integrals over a region (connected subset) of a 2D and a 3D manifold, where we add up the contributions ...
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### Question on equation of dimensions of tangent spaces in Proposition 5.38 of Lee's Smooth Manifolds

I am having trouble figuring out an equality on dimensions in the proof below from John Lee's Introduction to Smooth Manifolds. In the proof, how do we get $\dim T_p M - \dim T_{\Phi(p)}N=\dim T_p S$? ...
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### A regular, connected, compact surface with curvature on $[0,1]$

today was my final differential geometry exam and there was a problem that I partially solved, but I have some doubts. The problem asked to prove that there exists a regular, connected, compact ...
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### Characterization of the tangent space of an embedded submanifold via defining map

I have a confusion on function composition in the proof below by John Lee. We define $\Phi: U \to N$ to be a local defining map for $S$, i.e. $U$ is an open subset of $M$ and $S \cap U$ is a regular ...
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### On the proof of the uniqueness of smooth structures on submanifolds by John Lee

I am having difficulty with an argument in the proof below. So we assume $\tilde{S}$ to be an immersed submanifold with some topology other than the subspace topology and some smooth structure, and $S$...
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### $\frac{\partial }{\partial t}(f \circ \phi^{-1} \circ \phi \circ \gamma)(0)=\frac{\partial}{\partial t}(f \circ \psi^{-1} \circ\psi\circ \gamma)(0)$

I feel like this result is in a way obvious, since we can just apply the chain rule several times: \begin{aligned} \frac{d}{dt}(f \circ \phi^{-1} \circ \phi \circ \gamma)(0) &= D_{f\circ\phi^{-1}\...
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### Proving that the collection of PDFs using GMM will form a statistical manifold

If I have a set $X=\{x_1,x_2,...,x_n:x_{i}\in\mathbb{R^3};x_{i}\neq x_{j} \text{ for }i\neq j\}$. and suppose I use the Gaussian mixture model to map fit the probability density function to this $X$ ...
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### irreducible compoenets in Hamiton's first Ricci flow paper [closed]

In "Three-manifolds with positive Ricci curvature",page 288,11.6 lemma. I dont konw why $\left\lvert E_{ijk} \right\rvert ^2=\frac{7}{20}\left\lvert \partial_i R\right\rvert ^2$ ,I have used ...
1 vote
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### Optimizing linear least square subjected to manifold constraints

I am trying to optimize a function of a matrix $X \in \mathbb{R}^{m \times n}$ $$f(X) = \frac{1}{2} \left\lVert AX - B \right\rVert_F^2$$ s.t. $X^T X = I_n$ (identity matrix). As I've been watching ...
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### Fundamental group of manifold and axiom of choice

It is known that the fundamental group of a manifold must be countable. As far as I am aware, a proof of this uses in an essential way the Axiom of Countable Choice. I was wondering: is it consistent ...
I am reading Linear algebra and analysis by André Lichnerowicz and there is a notion on which I find nothing on the internet. First the context : We consider $E$ and $F$ real Euclidean spaces of ...