Manifolds Definition and 282 Threads

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of...

Given a topological manifold, this may or may not admit a ##C^1## atlas (i.e. starting from its maximal atlas it is or it isn't possible to rip charts from it to get an atlas of ##C^1## compatible charts). A theorem due to Whitney states that from such a topological manifold ##C^1##-atlas (if...

I've a doubt regarding Lemma 1.35 (Smooth Manifold Chart Lemma) from J. Lee "Introduction to Smooth Manifolds" The proof claims that Hausdorff property follows from v). However v) includes the case where both ##p## and ##q## are included in the same ##U_{\alpha}##, i.e. their neighborhoods are...

TL;DR Summary: I have a bad copy of book "On manifolds with an affine connection and the theory of general relativity"by Elie Cartan. Pages 36, 142 and 174 are partiy missing. I have a bad copy of Elie Cartan's book "On manifolds with an affine connection and the theory of general relativity"...

From my understanding, a topological manifold ##M## comes with, by definition, a locally euclidean topology and a (topological) atlas ##\mathcal A_1##. From this atlas one can construct the maximal atlas ##\mathcal A## throwing in all the chart maps ##(U,\varphi)## each from one of the open...

Suppose there was a bijection ##\varphi## between the 2-sphere ##M## and the euclidean plane ##\mathbb R^2##. Then one could endow ##M## with the initial topology from ##\mathbb R^2## through ##\varphi## turning it into an homeomorphism (this topology on ##M## would be different from the subset...

Consider a non-injective map ##\pi## from a set ##M## to a set ##N##. ##N## is equipped with a topological manifold structure (Hausdorff, second-countable, locally euclidean). Take the initial topology on ##M## given from ##\pi## (i.e. a set in ##M## is open iff it is the preimage under ##\pi##...

Consider the following: suppose there is a smooth vector field ##X## defined on a manifold ##M##. Take a smooth curve ##\alpha(\tau)## between two different integral curves of ##X## where ##\tau## is a parameter along it. Let ##A## and ##B## the ##\alpha(\tau)## 's intersection points with the...

Hi, I've a doubt about the following example in "Introduction to Manifold" by L. Tu. My understanding is that if one assumes the subspace topology from ##\mathbb R^2## for the "cross", then one can show that the topological space one gets is Hausdorff, second countable but non locally...

Hi, consider the group ##SL(n,\mathbb R)##. It is a subgroup of ##GL(n,\mathbb R)##. To show it is a Lie group we must assign a differential structure turning it into a differential manifold, proving further that multiplication and taking the inverse are actually smooth maps. With the...

Hi, in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space ##\mathbb R^3##. On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts. I believe the same view might also be...

Does the interior coordinate patch of the Schwarzschild analytic extension really describe the interior of a black hole? After all, that portion would have mass. Also, is there a way to describe just a black hole’s with regular spherical coordinates?

As discussed in a recent thread, I'd ask whether any smooth section over a Mobius strip must necessarily take value zero on some point over the base space ##\mathbb S^1##. Edit: my doubt is that any closed curve going in circle two times around the strip is not actually a section at all. Thanks.

Hi, I was keep reading the interesting book Exploring Black Holes - second edition from Taylor, Wheeler, Bertschinger. I'd like to better understand some points they made. In Box 3 section 3-6 an example of coordinate singularity at point O in Euclidean plane in polar coordinates centered there...

In differential geometry, we typically define the boundary ##\partial M## of a manifold ##M## as all ##p \in M## for which there exists a chart ##(U,\varphi), p \in U## such that ##\varphi(p) \in \partial\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n = 0 \}##. Consequently, we also demand that...

Hi, consider the set of the following parametrized matrices $$ \begin{bmatrix} 1+a & b \\ c & \frac {1 + bc} {1 + a} \\ \end{bmatrix} $$ They are member of the group ##SL(2,\mathbb R)## (indeed their determinant is 1). The group itself is homemorphic to a quadric in ##\mathbb R^4##. I believe...

Definitions: 1. A map ##p : X → Y## of smooth manifolds is called a trivial fibration with fiber ##Z## which is also a smooth manifold, if there is a diffeomorphism ##θ : X → Y ×Z## such that ##p## is the composition of ##θ## with the natural projection ##pr_1:Y × Z → Y##. 2. A map ##p: X →Y##...

Given the definition of a smooth map as follows: A continuous map ##f : X → Y## is smooth if for any pair of charts ##\phi : U →R^m, \psi:V →R^n## with ##U ⊂ X, V ⊂Y##, the map ##\phi(U ∩f^{-1}(V)) → R^n## given by the composition $$\psi ◦ f ◦ \phi^{-1}$$ is smooth. Claim: A map ##f : X → Y##...

Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold? Thank you!

Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$. I need to prove that the map is differentiable. But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable? MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math

Summary:: Books about Kähler manifolds, Ricci flatness and similar things Hi, I have an MSc degree in electronics, and I worked with IT. I was always interested in theoretical physics and did extra studies of this in my youth. Now as a pensioner I do private studies for fun. I have studied...

Hello! According to the attached proposition on ##C^\infty## manifold space of derivations ##D_m M## is isomorphic to Tangent space ##T_m M##. Cited here another proposition (1.4.5) states the following 1. For constant function ##D_m(f)=0## 2. If ##f\vert_U=g\vert_U## for some neighborhood...

Hi, I don't know if it is the right place to ask for the following: I was thinking about the difference between the notion of spacetime as 4D Lorentzian manifold and the thermodynamic state space. To me the spacetime as manifold makes sense from an 'intrinsic' point of view (let me say all the...

I'm having trouble with Rovelli's new book, partly because the info in it is pretty condensed, but also because his subjects are often very different from those in other books on GR like the one by Schutz. For one thing, he never uses the term "manifold", but talks about frame fields, which seem...

According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is: ##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0## Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...

Hi. I have a question. Two manifolds can be equivalent but have different co-ordinates and a correspondingly different metric defined on them. Is there an easy way to identify when two such manifolds are equivalent but just have different co-ordinates? What do I mean by...

Background: One can construct a Hilbert space "over" ##\mathbb{R}^{3}## by considering the set of square integrable functions ##\int_{\mathbb{R}^{3}}\left|\psi(\mathbf{r})\right|^{2}<\infty##. That's what is done in QM, and there, even if they are not normalizable, to every...

First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)

I was trying to look for something that works a lot of examples of integrals over surfaces, volumes etc. in general relativity. Tong's notes and some others are good on the abstract/theoretical side but it'd really be better at this stage to get some practice with concrete examples in order to...

I think it could be interesting. Consider a mechanical system A circle of mass M can rotate about the vertical axis. The angle of rotation is coordinated by the angle ##\psi##. A bead of mass m>0 can slide along this circle. The position of the bead relative the circle is given by the angle...

Hello. Why do we have different ways of determining curvature on manifolds like the sectional curvature, the scalar curvature, the Riemann curvature tensor , the Ricci curvature? What are their different uses on manifolds? Do they allow each of them different applications on manifolds? Thank you.

Summary:: hodge Duality... Does anybody has idea for solving these 2 problems?

Hello there. Do you know any examples of manifolds not being groups?Can you talk about some of them developing them as much as you want?Thank you.

Hello. Could we define a topological space that locally resembles a riemannian manifold or another manifold like a complex manifold, or a Hermitian manifold near each point? Could it have interesting properties and theorems? Thank you.

I was looking at this video , and I was wondering if a (Riemannian)manifold violates the "lorentz invariance" would it become a discrete manifold?

Hi, I just finished up with Riemann Geometry not to long ago, and did something with complex geometry on kahler manifolds. In your opinion what would be a next logical step for someone to study? I am very interested in manifold theory and differential geometry in general. I'm somewhat familiar...

Hi, I have just finished studying Riemannian Geometry and was moving on to trying to figure out what a Kahler manifold is. Using wikipedia's definition(probably a bad idea to start with) it says "Equivalently, there is a complex structure J on the tangent space of X at each point (that is, a...

[Moderator's note: Spin-off from another thread.] You need the structure of a topological vector field K with 0 as a limit point of K-{0}. The TVF structure allows the addition and quotient expression to make sense; you need 0 as a limit point to define the limit as h-->0 and the topology to...

I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...

Every two semi-Riemannian manifolds of the same dimension, index and constant curvature are locally isometric. If they are also diffeomorphic, are they also isometric?

Hi I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This...

Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?

Please forgive any confusion, I am not well acquainted with topological analysis and differential geometry, and I'm a novice with regards to this topic. According to this theorem (I don't know the name for it), we cannot embed an n-dimensional space in an m-dimensional space, where n>m, without...

Hello, In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as $$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}')...

Consider two pseudo-Riemmannian manifolds, ##M## and ##N##. Suppose that in coordinates ##y^\mu## on ##M## and ##x^\mu## on ##N##, the Riemann curvatures ##R^M## and ##R^N## of ##M## and ##N## are related by a coordinate transformation ##y = y(x)##: \begin{equation*} R^N_{\rho\mu\sigma\nu} =...

Hi there all, I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...

In the exercises on differential forms I often find expressions such as $$ \omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz $$ but this is only correct if we're in "flat" space, right? In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...

Let ##M## be an ##n##-dimensional (smooth) manifold and ##(U,\phi)## a chart for it. Then ##\phi## is a function from an open of ##M## to an open of ##\mathbb{R}^n##. The book I'm reading claims that coordinates, say, ##x^1,\ldots,x^n## are not really functions from ##U## to ##\mathbb{R}##, but...

Is it possible for a manifold to be homeomorphic to ##R^m## in some regions and homeomorphic to ##R^n## in other regions, with ##m \neq n##?

I'm reading "The Geometry of Physics" by Frankel. Exercise 1.3(1) asks what would be wrong in defining ##||X||## in an ##M^n## by $$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to...