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.
From the things I've studied till now, the thought came into my mind that all of the known solutions of Einstein's Field Equations are Lorentzian. Is it correct?
And if it is correct, is there something in EFEs that implies all solutions to it should be Lorentzian?
And the last question, do more...
I've been searching high and low through the Google for a solutions manual to William Boothby's "An Introduction to Differentiable Manifolds and Riemannian Geometry" to no avail. Does anyone know if ∃ such a thing? Thanks.
I have one question about Spivak's Calculus on Manifolds book. I have not learned directional derivatives and understand that these are left as exercises in his book, which would make one think these are not that important whereas he focuses on total derivatives or what you may name them...
I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I...
Hello I'm french so sorry for the mistake. If we have a manifold and a point p with a card (U, x) defined on on an open set U which contain p, of the manifold, we can defined the tangent space in p by the following equivalence relation : if we have 2 parametered curve : dfinded from...
Is the fact that all manifolds are hausdorff spaces a part of the definition, or can this be proven from the fact that it is a set which is locally isomorphic to open subsets of a hausdorff space?
P.S. if it can be proven I don't want to know the proof, I want to keep working on it, I just...
Hi, there is a result that every closed, oriented 3-manifold is the boundary of a 4-manifold that has only 0- and 2- handles. Anyone know other of these "boundary results" for some higher-dimensional manifolds, e.g., every closed, oriented k-manifold is the boundary of a (k+1)- dimensional...
Hi, we know that every contact manifold has a symplectic submanifold. Is it know whether every symplectic manifold has a contact submanifold?
A contact manifold is a manifold that admits a (say global) contact form: a nowhere-integrable form/distribution (as in Frobenius' theorem) ## w## so...
What can a complex manifold of dimension N do for me that real manifolds of dimension 2N can't.
Edit, I guess the list might be long but consider only the main features.
Thanks for any help or pointers!
Suppose that we have this metric and want to find null paths:
ds^2=-dt^2+dx^2
We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$
dx=\pm dt \to x=\pm t
Now switch to the more abstract and rigorous one-forms in differentiable manifolds.
Here \mathrm{d}t (v)...
Suppose that we have this metric and want to find null paths:
ds^2=-dt^2+dx^2
We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$
dx=\pm dt \to x=\pm t
Now switch to the more abstract and rigorous one-forms in differentiable manifolds.
Here \mathrm{d}t (v)...
Hi. I have a physics background but I am trying to get to grips with differential geometry and struggling with the abstract nature of it. I have a few questions if anyone can help ?
Is a smooth manifold the same as a differentiable manifold ? Does it have to be infinitely differentiable ? Is 3-D...
Is anyone familiar with this book?
Differentiable Manifolds: A Theoretical Physics Approach
https://www.amazon.com/gp/aw/s//ref=mw_dp_a_s?ie=UTF8&k=Gerardo+F.+Torres+del+Castillo&i=books&tag=pfamazon01-20
https://www.amazon.com/gp/product/0817682708/?tag=pfamazon01-20
If you are, what's your...
I am a graduate student in physics. One of my biggest frustrations in my education is that I often find that my mathematical background is lacking for the work I do. Sure I can make calculations adequately, well enough to even do well in my courses, but I don't feel like I really understand...
I am currently reading the paper given here by Acharya+Gukov titled "M Theory and Singularities of Exceptional Holonomy Manifolds", and in particular right now am following section 4 where the field content of the effective 4-dimensional theory is derived by harmonic decomposition of the 11D...
Hello,
I'm reading the book Geometrical methods of mathematial physics by Brian Schutz. In chapter 3, on Lie groups, he states and proves that the vector fields on a manifold over which a particular tensor is invariant (i.e. has 0 Lie derivative over) form a Lie algebra. And associated with...
Consider a smooth map ##F: M \to N## between two smooth manifolds ##M## and ##N##. If the pushforward ##F_*: T_pM \to T_{F(p)} N## is injective and ##F## is a homeomorphism onto ##F(M)## we say that ##F## is a smooth embedding.
In analogy with a topological embedding being defined as a map...
In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads:QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$.
Let $M$ be the set of all the points $\mathbf x$ such that $f(\mathbf x)=\mathbf 0$ and $N$ be the set of all the points $\mathbf x$ such...
As far as I have understood it submersions play the analogous role in manifold theory to quotient spaces in topology. Now is that suppose that we have submersion ##\pi: M \to N## with ##M## having a certain differential structure, then how is the differential structure of ##N## related to that...
Let ##M## and ##N## be smooth manifolds and let ##F:M \to N## be a smooth map. Iff ##(U,\phi)## is a chart on ##M## and ##(V,\psi)## is a chart on ##N## then the coordinate representation of ##F## is given by ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \to \psi(V)##. My question is...
Hi, I'm trying to show that if ## M^n ## is orientable and connected, with boundary (say with just one boundary component), then its top homology is zero. Sorry, I have not done much differential topology/geometry in a while.
I'm trying to avoid using Mayer-Vietoris, by using this argument...
Hello,
I am a mechanical engineer and I am teaching my self the topic of this subject line.
I now have a working understanding of the following: manifolds, exterior algebra, wedge product and some other issues. (I give you this and the next sentence so I can CONTEXTUALIZE my question.) I...
Hello,
I understand the concepts of real differentiable manifold, tangent space, atlas, charts and all that stuff. Now I would like to know how those concepts generalize in the case of a complex manifold.
First of all, what does a coordinate chart for a complex manifold look like? Is it a...
Hello,
I notice that most books on differential geometry introduce the definition of differentiable manifold by describing what I would regard as a differentiable manifold of class C∞ (i.e. a smooth manifold).
Why so?
Don't we simply need a class C1 differentiable manifold in order to...
Hello,
I was wondering if it is true that any open subset Ω in ℝn, to which we can associate an atlas with some coordinate charts, is always a manifold of dimension n (the same dimension of the parent space).
Or alternatively, is it possible to find a subset of ℝn that is open, but it is a...
Sorry guys, I have some differential topology homework, and I may be asking a lot of questions in the next few days.
Problem Statement
Suppose M_1,...,M_k are smooth manifolds and N is a smooth manifold with boundary. Then M_1×..×M_k×N is a smooth manifold with a boundary.
Attempt
Since...
In Sean Carroll's general relativity book he gives a requirement that two (differentiable) manifolds be the same manifold that there exist a diffeomorphism ##\phi## between them; i.e. a one-to-one, invertible and ##C^{\infty}## map.
Now I wanted to get some intuition why this is the best...
Consider the system $$x' = -x,$$ $$y' = y + g(x),$$ where $g$ is a class $C^1$ function with $g(0) = 0$.
Compute the stable manifold $W^s (\mathbf{0}).$
Using $g(x) = x^n (n \geq 1)$, compute $W^s (\mathbf{0})$ and $W^u (\mathbf{0})$.
The other was an exercise I found, this is an actual...
Consider the system of differential equations
$$x' = 2x - e^y (2+y),$$ $$y' = -y.$$
Find the stable and unstable manifolds near the rest point.
I know that the stable manifold $W^s$ is a immersed surface in $\mathbb{R}^2$ with tangent space $E^s$ (the stable linear subspace). How can I...
I am independently working through the topology book called, "Introduction to Topology: Pure and Applied." I am currently in a chapter regarding manifolds. They attempt to show that a connected sum of a Torus and the Projective plane (T#P) is homeomorphic to the connected sum of a Klein Bottle...
Given a closed Riemannian manifold, a point P on it and a nonzero vector V in its tangent space, can you extend a geodesic in that direction of V indefinitely? I count looping back onto itself as "indefinitely".
The theorem I have in my book only guarantees that this is possible locally near P.
Except for the work of Torsten I am not aware of any paper which discusses these topics.
Some ideas:
1) all theories for QG I am aware of do either use manifolds and smootheness (string theory, geometrodynamics, shape dynamics, ...) or are constructed from them (LQG, CDT, ...)
2) in some...
If we have a manifold with a chart projected onto ##R^n## cartesian space and define a curve ##f(x^\mu(\lambda))=g(\lambda)## then we can write the identity
\frac{dg}{d\lambda} = \frac{dx^\mu}{d\lambda} \frac{\partial f}{\partial x^\mu}
in the operator form:
\frac{d}{d\lambda} =...
This question comes from trying to generalize something that it easy to see for surfaces.
Start with an oriented surface smoothly embedded in Euclidean space. The embedding determines two mappings of the unit tangent circle bundle into Euclideam space.
Given a unit length tangent vector,e, at...
Hello everyone,
I am currently reading 'Geometrical Methods of Mathematical Physics' by Bernard Schutz and I have some questions about manifolds. I'm fairly new to Differential Geometry so bear with me!
On P33 he states that 'manifolds need have no distance relation between points, we...
Homework Statement
Consider the map \Phi : ℝ4 \rightarrow ℝ2
defined by \Phi (x,y,s,t)=(x2+y, yx2+y2+s2+t2+y)
show that (0,1) is a regular value of \Phi and that the level set \Phi^{-1} is diffeomorphic to S2 (unit sphere)
Homework Equations
The Attempt at a Solution
So I...
Can unquantized fields be considered smooth curved abstract manifolds? Say free particle solutions of the Dirac equation or the Klein Gordon equation? Can quantized fields also be considered curved abstract manifolds?
Thanks for any help!
Hello fellow mathematicians!
I am currently confused about the mathematics of curved manifolds.
When introducing the affine connection on our bundle it seems to be an object totally determined by our coordinate chart.
But since we can compute objects describing the curvature from the...
I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better
Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p\inM
Proposition: TpF : TpM → TF(p) is...
Dear All,
I've been studying differential geometry for some time, but there is one thing I keep failing to understand. Perhaps you can help out (I think the question is quite simple):
Can I use Cartesian coordinates to cover a curved manifold? I.e., is there an atlas that only contains...
One can do calculus on a differentiable manifold, what does that mean? Does it mean you can use differential forms on the manifold, or that you can find tangent vectors, What is certified as "calculus on a manifold".
Author: Michael Spivak
Title: Calculus on Manifolds
Amazon link: https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20
Prerequisities: Rigorous Calculus
Level: Undergrad
Table of Contents:
Foreword
Preface
Functions on Euclidean Space
Norm and Inner Product
Subsets of Euclidean...
Author: John Lee
Title: Riemannian Manifolds: An Introduction to Curvature
Amazon link https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20
Prerequisities: "Introduction to Smooth Manifolds" by Lee seems like a prereq.
Level: Grad
Table of Contents:
Preface
What Is Curvature?
The...
Author: John Lee
Title: Introduction to Smooth Manifolds
Amazon link https://www.amazon.com/dp/0387954481/?tag=pfamazon01-20
Prerequisities: Topology, Linear algebra, Calculus 3. Some analysis wouldn't hurt either.
Level: Grad
Table of Contents:
Smooth Manifolds
Topological Manifolds...
Author: John M. Lee
Title: Introduction to Topological Manifolds
Amazon Link: https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20
Prerequisities: Real Analysis course involving epsilon-delta and preferebly metric spaces, group theory
Level: Grad students
Table of Contents:
Preface...
Ok, so this relates to my homework, but I really can't find an answer anywhere, so this is more of a general question. First off, what does a "chart" of a manifold look like? Is it a set, a function, a drawing, a table, what?! I have found so many things about charts, but nothing shows what...
I have been working through Spivak's fine book, but the part about differential forms and tangent spaces has left me confused.
In particular, Spivak defines the Tangent Space \mathbb R^n_p of \mathbb R^n at the point p as the set of tuples (p,x),x\in\mathbb R^n. Afterwards, Vector fields are...
Ok first of all I'd like to mention that I've searched the forum and didn't find anything similar, so hopefully this thread is not unwelcome...
Now as the title suggests, I am interested in a parallelization of the two concepts. Personally I like to introduce the idea of submanifolds prior to...