Manifolds Definition and 283 Threads

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!

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...

Homework Statement Let ##A## be open in ##\mathbb{R}^n##; let ##\omega## be a k-1 form in ##A##. Given ##v_1,...,v_k \in \mathbb{R}^n##, define ##h(x) = d\omega(x)((x;v_1),...,(x;v_k)),## ##g_j(x) = \omega (x)((x;v_1),...,\widehat{(x;v_j)},...,(x;v_k)),## where ##\hat{a}## means that the...

Hi all, I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition: Given ##\textbf{x} \in \mathbb{R}^n##, we define a tangent vector to ##\mathbb{R}^n## at ##\textbf{x}## to be a pair...

Homework Statement Let ##O(3)## denote the set of all orthogonal 3 by 3 matrices, considered as a subspace of ##\mathbb{R}^9##. (a) Define a ##C^\infty## ##f:\mathbb{R}^9 \rightarrow \mathbb{R}^6## such that ##O(3)## is the solution set of the equation ##f(x) = 0##. (b) Show that ##O(3)## is a...

Homework Statement Let M and N be orientable m- and n-manifolds, respectively. Prove that their product is an orientable (m+n)-manifold. Homework Equations An m-manifold M is orientable iff it has a nowhere vanishing m-form. The Attempt at a Solution I assume I would take nowhere...

I am looking for books that introduce the fundamentals of topology or manifolds. Not looking for proofs and rigor. Something that steps through fundamental theorems in the field, but gives conceptual explanations.

Dear Folks: In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries...

Homework Statement This problem is in Analysis on Manifolds by Munkres in section 25. R means the reals Suppose M \subset R^m and N \subset R^n be compact manifolds and let f: M \rightarrow R and g: N \rightarrow R be continuous functions. Show that \int_{M \times N} fg = [\int_M f] [...

Ok, so I don't have much of an intuition for frame bundles, so I have some basic questions. A frame bundle over a manifold M is a principle bundle who's fibers are the sets of ordered bases for the vector fields on M right. 1) This means that any point in the fiber (say, over a point m in M)...

hi! i would very much like to have the "Lectures on Comlex Manifolds" by Philip Candelas. It was recommended by an instructor of a course on complex geometry i took some time ago, but sadly its out of print and not in our library. Does anyone has this documents in electronic form?

When reading other threads, following question crept into my mind: When given a manifold, why shouldn't I give it distance function by giving it a simple metric function, that is MxM→ℝ with the usual axioms? I could happily measure distances in coordinate-independent way for ever after...

Hi everebody, I want to clear something.An n-dimentional differential manifoled is locally endowed by topologies defined by the metrices from the local parametrisations.I suppose that these topologies may all be different.Am i right?If i am mistaken ,then why? thank's

I've always assumed that for a non-Euclidean manifold to exist, it has to be ambient in some higher-dimensional Euclidean space, like how a 2-sphere is ambient in 3-dimensional Euclidean space. But I've been hearing hints that higher-dimensional embedding is in fact unnecessary to define a...

If you have a strip and you bring it around so that the ends join, that is a manifold, call it X for convenience. If instead, you put a single twist in it before joining the ends, that is a Mobius strip, which is not homeomorphic to X. If you instead put two twists in it before you join the...

Hi, All: AFAIK, every complex manifold can be given a symplectic structure, by using w:=dz/\dz^ , where dz^ is the conjugate of dz, i.e., this form is closed, and symplectic. Still, I think the opposite is not true, i.e., not every symplectic manifold can be given a complex...

Hi, I want to ask a problem from Lee ' s book Introduction to Smooth Manifolds: Let F_p denote the subspace of C^\inf(M) consisting of smooth functions that vanish at p and let F^2_p be the subspace of F_p spanned by functions fg for some f,g \in F_p. Define a map \phi: F_p----->...

Hi, I'm having trouble understanding why is tangent space at point p on a smooth manifold, not embedded in any ambient euclidean sapce, has to be defined as, for example, set of all directional derivatives at that point. To my understanding, the goal of defining tangent space is to provide...

If we consider a Riemannian surface as a one-dimensional complex manifold, what does that tell us about its intrinsic curvature? I mean for one-dimensional curves we know they only have extrinsic curvature so it depends on the embedding space, this doesn't seem to be the case for one-dimensional...

Hi! I want to know if any smooth manifold in n-dimensional euclidean space can be compact or not. If it is possible, then could you give me an example about that? I also want to comfirm whether a cylinder having finite volume in 3-dimensional euclidean space can be a smooth manifold. I...

Consider the infinite disjoint union M = \coprod\limits_{i = 1}^\infty {M_i },where M_i 's are all manifolds of finite type of the same dimension n.Then the de Rham cohomology is a direct product H^q (M) = \prod\limits_i {H^q (M_i )}([SIZE="5"]why?),but the compact cohomology is a direct sum...

This is really just a general question of interest: can every smooth n-manifold be embedded (in R2n+1 say) so that it coincides with an affine variety over R? Does anyone know of any results on this?

Does anyone know what the Asian characters on this book mean? Why are they there?

I am having some trouble understanding the notion of an orientated manifold. But first let me get some preliminary definitions out of the way: A diffeomorphism is said to be orientation-preserving if the determinant of its Jacobian is positive. A k-manifold M in \mathbb{R}^n is said to be...

Background: I'm going to be a junior, having very strong Analysis and Algebra yearlong sequences, in addition to a very intense Topology class, and a graduate Dynamical Systems class. For this coming Fall, I'm sort-of registered for this class, titled "Manifolds and Topology I" (part of a...

Right now, I'm reading through Lee's Intro to Smooth Manifolds and I was wondering if there is a website somewhere that has problems different from the book. Or if there is another book out there that covers about the same material as Lee's that would be good to. Thanks in advance.

Hi, All: I am trying to show that the connected sum of orientable manifolds M,M' is orientable , i.e., can be given an orientation. I am using the perspective from Simplicial Homology. Consider the perspective of simplicial homology, for orientable manifolds M,M', glued about cycles C,C'...

Hi, All: Given manifolds M,N (both embedded in $R^n$, intersecting each other transversally, so that their intersection has dimension >=1 ( i.e. n -(Dim(M)-Dim(N)>1) is the intersection a manifold? Thanks.

I am currently going through some online notes on differential geometry and general relativity. So far I have been following pretty well until I got to 3.4 cont'd letter (e) of http://people.hofstra.edu/Stefan_Waner/diff_geom/Sec3.html" document and the definition directly after. My first...

My definition of diffeomorphism is a one-to-one mapping f:U->V, such that f and f^{-1} are both continuously differentiable. Now, how to prove that if f is a diffeomorphism between euclidean sets U and V, then U and V must be in spaces with equal dimension (using the implicit function theorem)?

Quoted from a book I'm reading: if f is any function defined on a manifold M with values in Banach space, then f is differentiable if and only if it is differentiable as a map of manifolds. what does it mean by 'differentiable as a map of manifolds'?

im trying to read calculus on manifolds by michael spivak and am having a tough time with it. if anyone could recommend a more accessible book (perhaps one with solved problems) id really appreciate it.

What yould you answer if a professor asks you, Why are the classification theorems of manifolds so important? Why was the classification of surfaces celebrated?

Hi, where I can find a proof of the theorem that establishes that a manifold is metrizable (with a riemannian metric) if and only if is paracompact?.

Hi everyone, been away for a while I got bogged down with my classes so didn't have time to work on this book and haven't been on the forums much. Was getting caught back up to where I was before in here and I ran into a problem that I can't figure out the notation on. I am only looking for...

hi friends :) is there someone who has studied the spectrum of a Riemannian Laplacian? I have a question on this subject. Thank you very much for answering me.

Hii all :smile: What is Fuzzy manifold ? and what is deformation ? thank u >>

A little embarrassing, but I have had very little exposure to anything involving manifolds and am trying to work through these notes over spring break. I will have many questions on even the simplest concepts. In this thread I hope to outline these as I encounter them, and if anyone can help I...

my book defines an orientation preserving parametrization of a manifold as one such that: Ω(D1γ(u), ..., Dkγ(u)) = +1 for all u in the domain of γ, where D1,...Dk are the derivatives of the parametrization γ. my book also defines the orientation of a surface in R^3 by Ω(v1,v2) = sgn...

Homework Statement Taken from Wiki: a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus, a line and a circle are one-dimensional manifolds, a plane and sphere (the surface of a ball)...

I understand that symplectic manifolds are phase spaces in classical mechanics, I just don't understand why we would use them. I understand both the mathematics and the physics here, it is the connection between these areas that is cloudy... What on Earth does the symplectic form have to do...