Tagged Questions

1
vote
0answers
17 views

Lebesgue covering dimension of a manifold

I have found many sources saying that the Lebesgue covering dimension of a (topological or smooth) manifold is the same as the dimension of the manifold. Does anyone know where I can find the proof?
4
votes
0answers
50 views

Differentiable manifolds, Serge Lang

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
1
vote
0answers
22 views

Standards in P.L. Topology

About a week ago, the reading course on PL topology I'm going to follow started. The aim of the reading course is to understand the basics of PL topology and have a reasonable to good understanding of ...
8
votes
0answers
165 views

Infinite dimensional constant rank theorem

Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
0
votes
1answer
38 views

Metric Spaces needed for Differential Geometry

I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and ...
2
votes
0answers
63 views

Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd). Questions: Suppose that we ...
4
votes
1answer
116 views

Green's function for the Yamabe problem

I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8. Theorem 2.8 (Existence of the Green Function). Suppose $M$ is a ...
2
votes
1answer
58 views

Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations

Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$ Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on ...
3
votes
2answers
153 views

Learning about Manifolds

I am looking to learn about manifolds for use in signal processing. I have a engineering degree where I have covered calculus and basic linear algebra, with this background in mind, does anyone have a ...
2
votes
1answer
158 views

Book recomendations for Smooth manifolds.

I want to learn about smooth manifolds, I have never studied them before, but I have a good background in Algebra. Can any one recomend some good introductory books? Thanks
0
votes
3answers
165 views

Simpler definition of manifold

I'm new to topology but I must understand how it works to progress in my research. First, can anybody point me to a document that introduces topology in a "gentle" way. What pre-requisites do I need ...
4
votes
0answers
115 views

de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...
3
votes
3answers
245 views

Prerequisites for studying smooth manifold theory?

I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to ...
4
votes
1answer
111 views

$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of ...
5
votes
1answer
99 views

Status of PL topology

I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological ...
10
votes
4answers
753 views

Reference on Geometric Topology

Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional ...
1
vote
1answer
166 views

The Implicit Function Theorem for complex polynomials

I'm looking for a reference that proves implicit function theorem for polynomials in two variables over the complex numbers via the real version. Such a theorem is needed, for example, in the theory ...
1
vote
0answers
51 views

Fundamental Domain of Manifold Reference Request

I am interested in learning about the fundamental domain of a manifold and I am wondering if anyone know of any papers or descriptions online other than Wikipedia and the linked articles? I am looking ...
2
votes
0answers
113 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
49
votes
5answers
1k views

Defining a manifold without reference to the reals

The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. ...
2
votes
0answers
112 views

How to prove that a certain action is hamiltonian?

Reading a paper I had the need to complete a proof, and come up with a certain argument(see below). My question is: could I reduce it to a special case of some theorem? I ask this question in order to ...
7
votes
2answers
143 views

Demonstrating the value of abstracting away from elements/subsets to maps

Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction: an actual element, e.g. $s\in S$ an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$ an ...
4
votes
7answers
2k views

Good introductory book on Calculus on Manifolds

I have already taken up to Multivariable Calculus, Linear Algebra and Diff Eq. I want to learn Calculus on Manifolds by myself, could you recommend a good introductory book on this subject? Should I ...