In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.
One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics and augmented-reality given the need to associate pictures (texture) to coordinates (e.g. CT scans).
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
The study of manifolds requires working knowledge of calculus and topology.
I am applying a Green's probabilistic elastodynamic tensor with relativistic manifold extensions to solve a pull out of a smoothly shaped deformable spheroid from a stiff inhomogenous deformable quasi-brittle host. This involves a Hooke's law tensor, a relativistic manifold Ricci tensor, a...
On page 30 of the notes (https://arxiv.org/abs/1501.00007) by Veronika Hubeny on The AdS/CFT correspondence, we find the following:
So far, we have been describing just one particular case of the AdS/CFT duality, namely (3.3). There are however many ways in which the correspondence can be...
Hi
I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...
Any open subset of ##\mathbb{R}^{n}##;
The n-Sphere, ##\mathbb{S}^n##;
The Klein Bottle.
I guess they don't have a boundary, as a neighborhood of any point of them is homeomorphic to ##\mathbb{R}^n##.
I'd like to know whether my guess is correct and whether the reason I'm giving for them not to...
Why manifolds in General (and Special) Relativity have to be open? Would this be because an open manifold have a continuous interval? (i.e. an interval with no interruptions)
Are the metrics for say the Calabi-Yau manifolds of string theory, assuming they have a metric, dynamic in the sense that a vibrating string interacts with the compact space causing the metric to change where there is a string, even if only a tiny amount?
Thanks!
In my ignorance, when first learning, I just assumed that one pushed a vector forward to where a form lived and then they ate each other.
And I assumed one pulled a form back to where a vector lived (for the same reason).
But I see now this is idiotic: for one does the pullback and pushforward...
Manifolds of contant curvature are conformally flat. I'm trying to find a stronger claim related to this for manifolds of dimension >2. Does anyone knows if for instance Riemannian manifolds(of dimension >2) with non-constant curvature are necessarily not conformally flat, or maybe something...
Hello every one .
first of all consider the 2-dim. topological manifold case
My Question : is there any difference between
$$f \times g : R \times R \to R \times R$$
$$(x,y) \to (f(x),g(y))$$
and $$F : R^2 \to R^2$$
$$(x,y) \to (f(x,y),g(x,y))$$
Consider two topological...
So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω.
And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...
I'm interested in this subject. This is a graduate text and I believe the prereqs are mostly a math degree, which I somewhat have(B.S in Applied Math from a few years back). The thing is, I forgot details about things. For example, I know how to do an epsilon delta proof and can read one when...
1. The problem statement, all variables and given
Before I try to work through the book, it would
be great to have a list of typos, if there are
any.
Homework EquationsThe Attempt at a Solution
I would like to know at what level is the book Tensors and Manifolds by Wasserman is pitched and what are the prerequisites of this book? Given the prerequisites, at what level should it be (please give examples of books)? If anyone has used this book can you please kindly give your comments and...
I have a question regarding the usage of notation on problem 2-11.
Find ##f'(x, y)## where ## f(x,y) = \int ^{x + y} _{a} g = [h \circ (\pi _1 + \pi _2 )] (x, y)## where ##h = \int ^t _a g## and ##g : R \rightarrow R##
Since no differential is given, what exactly are we integrating with...
While Minkowski space and Euclidean space both have identically zero curvature tensors it seems that a flat Lorentz manifold in general, may not admit a flat Riemannian metric. Such a manifold is the quotient of Minkowski space by the action of a properly discontinuous group of Lorentz...
In my introduction to manifolds the following is stated:
Polar coordinates (r, phi) cover the coordinate neighborhood (r > 0, 0 < phi < 2pi); one
needs at least two such coordinate neighborhoods to cover R2.
I do not understand why two are needed. Any point in R2 can be described by polar...
I am really struggling with one concept in my study of differential geometry where there seems to be a conflict among different textbooks. To set up the question, let M be a manifold and let (U, φ) be a chart. Now suppose we have a curve γ:(-ε,+ε) → M such that γ(t)=0 at a ∈ M. Suppose further...
As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...
I'm fairly new to differential geometry (learning with a view to understanding general relativity at a deeper level) and hoping I can clear up some questions I have about coordinate charts on manifolds.
Is the reason why one can't construct global coordinate charts on manifolds in general...
I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the...
I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation).
Later, I run into the...
No question this time. Just a simple THANK YOU
For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups.
My math background was very deficient: I am a 55 year old retired (a good life) professor of...
I have found the following entry on his blog by Terence Tao about embeddings of compact manifolds into Euclidean space (Whitney, Nash). It contains the theorems and (sketches of) proofs. Since it is rather short some of you might be interested in.
I have found this paper on the internet and think it might be interesting for some on this forum because there are frequently questions similar to the ones the paper tries to answer.
http://arxiv.org/abs/1605.00890
http://arxiv.org/pdf/1605.00890v1.pdf
Hello,
As you might discern from previous posts, I have been teaching myself:
Calculus on manifolds
Differential forms
Lie Algebra, Group
Push forward, pull back.
I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...
This is from Jackson, "Electrodynamics"
a field is a fuction mapping phi: M -> T, x -> phi(x) from a base manifold M into a target manifold T.
field X: R3 * R1 -> R3
x(r,t) ->X(x)I think this is eucledian R4 to R3 so I wonder why Jackson explained this with the concept of manifolds?
Is it...
Hello! Good morning to all forum members!
I am studying general relativity through the wonderful book: "General Relativity: An Introduction for Physicists" by M.P. Hobson (Cambridge University Press) (2006). My question is about Riemannian manifolds and local cartesian coordinates (Chapter 02 -...
I am reading John M. Lee's book: Introduction to Smooth Manifolds ...
I am focused on Chapter 1: Smooth Manifolds ...
I need some help in fully understanding Example 1.3: Projective Spaces ... ...
Example 1.3 reads as follows:My questions are as follows:Question 1In the above example, we...
I am reading "An Introduction to Differential Topology" by Dennis Barden and Charles Thomas ...
I am focussed on Chapter 1: Differential Manifolds and Differentiable Maps ...
I need some help and clarification on an apparently simple notational issue regarding the definition of a chart...
As many of you know, using the stereographic projection one can construct a homeomorphism between the the complex plane ℂ1 and the unit sphere S2∈ℝ3. But the stereographic projection can be extended to
the n-sphere/n-dimensional Euclidean space ∀n≥1. Now what I am talking about is the the...
Does the words "Quantum Manifolds" make any sense?
Can quantum occurs on a manifold?
Or do you automatically equate manifolds to lorentzian manifold and it becomes a problem of quantum gravity?
Or can there be manifolds not related to spacetime.. so can quantum on manifold make sense?
Hi,
I am just about to finish working through the integration chapter of calculus on manifolds, and I am wondering whether it would be better to get spivaks first volume of differential geometry (or another book, recommendations?) and start on that, or to finish calculus on manifolds first...
Dear Physics Forum friends,
I am currently trying to purchase Munkres' Analysis on Manifolds to replace the vector-calculus chapters of Rudin-PMA, which is quite unreadable compared to his excellent chapters 1-8. I know that there is a paperback-edition for Munkres, but I heard that the...
Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...
I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the Ricci rotation coefficients...
Dear all
We all agree that a manifold is globally non euclidean but locally it is. So we can find near each point a hemeomorphic to an open set of euclidean space of the same dimension as the manifold. This is a general definition for all manifold to follow. Then what is the difference between...
Hi
How can a person chart a manifold if he does not know how the manifold looks like ?
E.g. The 2-sphere manifold can have 2 charts and symmetric charts with the chart goes like this ( theta from zero to pi , psy from minus pi to pi ) but the problem for unknown manifold , e.g. In general...
A spacetime is said to be time-orientable if a continuous designation of which timelike vectors are to be future/past-directed at each of its points and from point to point over the entire manifold. [Ref. Hawking and Israel (1979) page 225]
I want to make sure what conditions must hold in...
I know that manifolds are topological spaces that locally look like euclidean spaces near each point of and open neighbourhood
And non-euclidean spaces are the curved spaces or simply don't match the 5th euclid's axiomThanks .
Homework Statement
Let ##M## be the set of all points ##(x,y) \in \mathbb{R}^2## satisfying the equation
##xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 ##
Prove that ##M## is a manifold. What is the dimension of ##M##?
Homework EquationsThe Attempt at a Solution
I think this question it started...
I am trying to finish the last chapter of Spivak's Calculus on Manifolds book. I am stuck in trying to understand something that seems like it's supposed to be trivial but I can't figure it out.
Suppose M is a manifold and \omega is a p-form on M. If f: W \rightarrow \mathbb{R}^n is a...
Hello every one
Can one say , that
A globle coordinate chart is a cartesian coordinate
And a local coordinate chart is any kind of curvilinear coordinate ?Thanks
Greetings. I just bought a textbook and I have no idea what it is about. A little explanation is in order:
One of my goals in life has been to obtain a degree in mathematics. Unfortunately, I have made very poor life choices that have made this goal practically unachievable, which I won't...
So this is beginning to feel like the beginning of the 4th movement of Beethoven's Ninth: it is all coming together.
Manifolds,Lie Algebra, Lie Groups and Exterior Algebra.
And now I have another simple question that is more linguistic in nature.
What does one mean by "Calculus on Manifolds"...
I have the opportunity to pursue an independent study in functional analysis (using Kreyszig's book) or calculus on manifolds (using Tu's book) next semester. I think that both of the subjects are interesting and I would like to study them both at some point in my life, but I can only choose one...
In my book its says let i: U →M (but with a curved arrow) and calls it an inclusion map. What exactly is an inclusion map? Doesn't the curve arrow mean its 1-1? So are inclusion maps always 1-1?
In all the notes that I've found on differential geometry, when they introduce integration on manifolds it is always done with top forms with little or no explanation as to why (or any intuition). From what I've manage to gleam from it, one has to use top forms to unambiguously define...
Can anyone give a laymans explanation of conformal time in relativity? I tried to read Roger Penrose's book but I found it hard to grasp.Thanks in advance .
Also is a Lorentzian manifold different to a conformal manifold? A laymans explanation would also be much apprecitaed.
What is the motivation for defining vectors in terms of equivalence classes of curves? Is it just that the definition is coordinate independent and that the differential operators arising from such a definition satisfy the axioms of a vector space and thus are suitable candidates for forming...
Hello! :o
I am doing my master in the field Mathematics in Computer Science. I am having a dilemma whether to take the subject Partial differential equations- Theory of weak solutions or the subject differentiable manifolds.
Could you give me some information about these subjects...