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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Submanifold of the space of holomorphic functions

Let $H$ be the space of holomorphic functions from the unitary disk $\Delta \subseteq \mathbb{C}$ to $\mathbb{C}^2$ mapping $0 \in \mathbb{C}$ to $0 \in \mathbb{C}^2$. Let $x \in \mathbb{C}^2$ be a ...
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1 vote
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Is a volume defined in manifold with spacetime-signature?

A volume in $R^n$ is easily calculated by multiplying the lengths of the $n$ dimensions. I'm wondering how the different sign of time acts on the volume in a manifold with spacetime signature (1,3). ...
1 vote
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Constructing cut-off function on geodesic ball

Let $M$ be a complete non compact Riemannian manifold, $p \in M$ be a fixed point, $r > 0$ be an arbitrary (as large as one may want it to be) constant and $B_r(p) = \exp(B_r(0))$ the geodesic ball ...
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1 vote
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Can we know that a map is a submersion with using only its level sets

Let $M$ be a manifold of dimension $2n$, $f:M\rightarrow\mathbb{R}^n$ a smooth map. We know that if $f$ is a submersion, then for any $c\in f(M)$, $f^{-1}(c)$ is a $n$-submanifold of $M$. But what ...
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Is there manifold, whose cohomology or homology groups are Lie groups?

Is there manifold, whose cohomology or homology groups are Lie groups? I have scanned to introductory article https://en.wikipedia.org/wiki/Cohomology and the examples mentioned there does not include ...
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1 vote
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Coordinate patches attached to a 2-manifold

In Road to reality page-185, the following photo is shown: I am a bit confused here because I thought the charts of the manifold existed as flat sheets which are subsets of $R^2$ (for a 2- manifold ...
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Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
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1 vote
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Equivalent definitions of Sobolev space on manifold and references

It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$: D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm. D2. All functions ...
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1 vote
26 views

Riemannian distance via the length of a curve

I am reading a book where they define the Riemannian distance between two points on a manifold. Naturally it is given as the infimum of the integral of the length of the curves which connect the two ...
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Prerequisites for John M. Lee's Introduction to Topological Manifolds?

I currently have some questions regarding prerequisites, especially prerequisites on analysis before diving into that book. I really liked the way author explained things from some parts I read and I'...
1 vote
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Relation between mod 2 Betti numbers and integral cohomology

Let $M$ be an orientable connected manifold of finite type. My question is that if $H^{i}(M,\mathbb{Z}_{2})$ is non-zero, then can we say that $H^{i}(M,\mathbb{Z})$ is non-zero? I know that this is ...
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Submersion, diffeomorphism, and embedding

Let $\pi:M \rightarrow N$ a submersion and $f:S \rightarrow M$ a smooth map such that $\pi \circ f : S \rightarrow N$ is a diffeomorphism. Show that $f:S \rightarrow M$ is an embedding. I'm having ...
1 vote
74 views

Why are closed manifolds defined as they are?

The standard definition of a closed manifold is the following: A closed manifold is a manifold without boundary that is compact. I am wondering, why we request that the manifold is compact. As I ...
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Quotients of real projective spaces

I am trying to determine whether or not $M := \mathbb{R}P^4/\mathbb{R}P^2$ is a manifold. I believe it is clear that $M$ is second-countable. However, I'm not not sure about Hausdorffness and being ...
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Verifying Stokes' Theorem with $\omega = x^2 \text{d}w-2yz \text{d}x$ as a one-form in $\mathbb{R}^4$ over a manifold.

I'm having trouble verifying Stokes' Theorem, and I'm just wondering if someone's able to find my mistake in the calculation. Let $\omega=x^2\text{d}w-2yz \text{d}x$ be a one-form on $\mathbb{R}^4$, ...
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Question $(I).$ Show that if the $n$-dimensional manifold $M$ is a product of spheres, then there exists an embedding $M \to \mathbb R^{n+1}.$ Question $(2).$ Show that there exists an embedding \$S^{...