Manifolds Definition and 290 Threads

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...

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.

Does anyone know of fields (with additional structure/properties) other than either ##\mathbb R, \mathbb C ## that are "naturally" manifolds? 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...

Author: James R. Munkres Title: Analysis On Manifolds Amazon Link: https://www.amazon.com/dp/0201315963/?tag=pfamazon01-20 Prerequisities: Contents:

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...