Articles of manifolds

Statistical Inference and Manifolds

I have just begun approaching the connection between statistical inference and differencial geometry. If I got it correctly, one of the most fundamental concept regards the connection between a $ P(x; \xi) $ Statistical Distribution with $ \xi $ Parameter Vector and a certain $ V $ Manifold for which $ \xi $ can be […]

The integral of a closed form along a closed curve is proportional to its winding number

Source: Guillemin-Pollack Exercise 4.8.2. Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 – \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 – \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where $W(\gamma, 0)$ is the winding number of $\gamma$ with respect to the origin. $W(\gamma, 0)$ is defined just like $W_2(\gamma, 0)$, but using […]

Free and proper action

I don’t know how to solve this problem. Let G be a Lie group and H a closed Lie subgroup ,that is, a subgroup of G which is also a closed submanifold of G. Show that the action of H in G defined by A(h,g)=h.g is free and proper. Can you please define what is […]

Existence of a universal cover of a manifold.

Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don’t come along with this, like a global Hausdorff condition mentioned here, second countability or paracompactness etc. Mainly it would seem to rule out certain pathological […]

Any two $1$-forms $\alpha$, $\alpha'$ with property satisfy $\alpha = f\alpha'$ for some smooth nowhere zero function $f$?

This is a followup question to here. Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$? Does it follow that any two $1$-forms $\alpha$, $\alpha’$ […]

Questions about manifolds

This is a question from spivak and “proper” means the inverse image of any compact set of N is still compact. However, I can not find a suitable compact subset of N to use this property.

Is this hierarchy of manifolds correct?

Question: a Hausdorff differentiable manifold (locally Euclidean space): $$ \text{is metrizable} \iff \text{is paracompact} \iff \text{admits a Riemannian metric} \,?$$ Does one also have for a locally Euclidean Hausdorff space (not necessarily differentiable): $$\text{second countable} \iff \text{metrizable with countably many connected components}? $$ Thus second countability is strictly stronger for such spaces than metrizability/paracompactness/existence of […]

Why $M=\{(x, |x|), x\in\mathbb{R}\}$ is not an embedded submanifold?

How can I prove that $M=\{(x, |x|), x\in\mathbb{R}\}$ is not an embedded smooth($C^\infty$) submanifold of $\mathbb{R}^2$. I tried to say there is any ($C^{\infty}$) immersion from $\mathbb{R}$ into $\mathbb{R}^2$ such that its image is $M$, but I couldn’t. I attend all of you to the following point( My friends!! please do not beguile for the […]

Unique manifold structure

I am reading the first chapter from the book – Foundations of Differentiable manifolds and Lie groups by Warner. There, he has given two statements to be proved as exercises. a) Let $M$ be a differentiable manifold and $A$ be a subset of $M$. Fix a topology on $A$. Then there is at most one […]

Pipe-fitting conditions in 3D

Let’s we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a tube of diameter $D$ around it. Questions: What are the set of conditions this curve has to satisfy to make sure tube is not self intersecting and surface area is still smooth (differential)? […]