Manifolds Definition and 283 Threads

Hello, I am learning about manifolds but I am not understanding part of the definition. This is what I'm looking at for defining the n-manifold M. (i) M is Hausdorff (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets I have a problem with (ii)...

Hi all, I was reading a paper written by Brian Greene sometime ago on flop transitions where one can essentially change the topology of the manifold but the four-dimensional physics that applied to the older manifold still holds. From that I am trying to extrapolate the following: Is it...

Hello there! I just started reading Topological manifolds by John Lee and got one questions regarding the material. I am thankful for any advice or answer! The criteria for being a topological manifold is that the space is second countantable ( = there exists a countable neighborhood...

I saw a nice formulation of the variation on odd dimensional manifolds in the paper of http://arxiv.org/abs/math-ph/0401046" : The referenced book of Arnold uses completely different formalism than this. I don't see clearly the connection between the traditional calculus of variations...

Hello, I read that when a manifold has a flat metric, the geodesics are always straight lines in the parameter space. I have two questions: (1) If we are given a Clifford torus S^1 \times S^1 (which is flat), how do we compute the geodesic between two points? Is the following correct...

I hope this is not the wrong place to ask this... Can anybody tell me if it is possible to find "Spivak calculus on manifolds" on line (a PDF copy for example) Thanks

This should hopefully be a quick and easy answer. I'm running through Lee's Introduction to Smooth Manifolds to brush up my differential geometry. I love this book, but I've come to something I'm not sure about. He states a result for which the proof is an exercise: I'm not quite clear on...

Hi, Everyone: A question on knots, please; comments,references appreciated. The main points of confusion are noted with a ***: 1)I am trying to understand how to describe the knot group Pi_1(S^3-K) as a handlebody ( this is not the Wirtinger presentation; this is from some...

Hello, it is known that pure-quaternions (scalar part equal to zero) identify the \mathcal{S}^2 sphere. Similarly unit-quaternions identify points on the \mathcal{S}^3 sphere. Now let's consider quaternions as elements of the Clifford algebra \mathcal{C}\ell_{0,2} and let's consider a...

Hey, I'm trying to do some optimization on a manifold. In particular, the manifold is \mathfrak U(N) , the NxN unitary matrices. Now currently, I'm looking at "descent directions" on the manifold. That is, let f: \mathfrak U(N) \to \mathbb R be a function that we want to minimize, p...

http://planetmath.org/?op=getobj&from=objects&id=4370 that's pretty much the proof of Stolkes Theorem given in Spivak but I'm having a lot of difficulty understanding the details specifically...when the piecewise function is defined for j>1 the integral is 0 and for j=1 the integral is...

I am confused about the different Ricci-named objects in complex and specifically Kahler geometry: We have the Ricci curvature tensor, which we get by contracting the holomorphic indices of the Riemann tensor. We have the Ricci scalar Ric, which we get by contracting the Ricci tensor. Then there...

Hi, everyone: I am trying to show that any complex manifold is orientable. I know this has to see with properties of Gl(n;C) (C complexes, of course.) ; specifically, with Gl(n;C) being connected (as a Lie Group.). Now this means that the determinant map must be either...

What's a really good resources with numerous easy-to-follow proofs to theorems on symplectic manifolds? Arnold is too difficult.

Homework Statement Let f: M \rightarrow N , g:N \rightarrow K , and h = g \circ f : M \rightarrow K . Show that h_{*} = g_{*} \circ f_{*} . Proof: Let M, N and K be manifolds and f and g be C^\infinity functions. Let p \in M. For any u \in F^{\infinity}(g(f((p))) and any...

Hi, I'm having some trouble understanding this theorem in Lang's book, (pp. 497) "Fundamentals of Differential Geometry." It goes as follows: \int_{M} \mathcal{L}_X(\Omega)= \int_{\partial M} \langle X, N \rangle \omega where N is the unit outward normal vector to \partial M , X...

The most recent version of the theorem, as stated by Nikonorov in 2004 Let G a connected compact semisimple Lie group, which acts almost effectively on a seven-dimensional simply-connected homogeneous space M^7=G/H. If (G/H,\rho) is a homogeneous Einstein manifold, then it is either a symmetric...

Why all two dimensional manifolds are conformally flat? Why all manifolds with constant sectional curvature are conformally flat? Does anyone know proofs of above statements. Thanks in advance.

I think I read on these forums that a small piece of relatively flat 4D spacetime of General Relativity can be embedded in ten dimensions. What happens if the small piece of spacetime is not very flat, does this change the number of embedding dimensions required? Thanks for any help!

Hello, I will expose a simplified version of my problem. Let's consider the following transformation of the x-axis (y=0) excluding the origin (x\neq 0): \begin{cases} \overline{x}=x \\ \overline{y}=1/x \end{cases} Now the x-axis (excluding the origin) has been transformed into an hyperbola...

What conditions do we have to put on a pseudo-Riemannian manifold in order for a unique and well defined concept of distance between events to be meaningful? I'm thinking about for example max. length of geodesic connecting two events. We have to require one such maximum or minimum length to...

So I'm trying to understand the statement: On a complex manifold with a hermitian metric the Levi-Civita connection on the real tangent space and the Chern connection on the holomorphic tangent space coincide iff the metric is Kahler. I basically understand the meaning of this statement, but...

I am studying calculus and statistics currently, and a possible relationship between them just occurred to me. I was thinking about two things: (i) is a differentiable function from R to R a manifold, and (ii) in what way is a random event really unpredictable? So I don't know much about either...

Hi there, Is there an "easy" way to find a tangent space at a specific point to an implicitly defined manifold? I am thinking of a manifold defined by all points x in R^k satisfying f(x) = c for some c in R^m. Sometimes I can find an explicit parametrization and compute the Jacobian matrix...

Show that U(x0, ε) is an open set. I'm reading Analysis on Manifolds by Munkres. This question is in the review on Topology section. And I've just recently been introduced to basic-basic topology from Principles of Mathematical Analysis by Rudin. I'm not really certain where to begin...

In string theory are the strings themselves the manifolds? or are the strings vibrating in a 10 or 11-d manifold?.

Seen a couple of pieces on gauge theories of finance equating arbitrage opportunities to curvature, for this to work the curvature must therefore not be constant, instead it would be some sort of a function of capital flows

Does every (continous and second countable) topological manifold have an Euclidean neighbourhood around each of its points whose closure equals the whole manifold?

The problem is to prove that if M and N are orientable manifolds, then MxN is an orientable manifold

[FONT="Arial"][SIZE="5"]Please: I need abook that include this Definitions: 1- Manifold 2- Stable Manifold 3- unstable Manifold thank you.

I've read that with 2D manifolds, you can create any closed 2D manifold by adding "handles" or "crosscaps" (or "crosshandles"). To add a handle, cut out two disks and add the ends of a circle x interval product (cylinder). If you glue one end "the wrong way" you get a "crosshandle", which is how...

I am currently working through spivak's calculus on manifolds (which i love by the way) in one of my class. my question is about his notation for partial derivatives. i completely understand why he uses it and how the classical notation has some ambiguity to it. however, i can't help but...

What are some preliminary texts/knowledge before approaching: Spivak's Calculus on Manifolds?

Hi, I'm confused about what differentiation on smooth manifolds means. I know that a vector field v on a manifold M is a function from C^{\infty}(M) to C^{\infty}(M) which is linear over R and satisfies the Leibniz law. This should be thought of, I'm told, as a 'derivation' on smooth...

What I'm about to say is really sketchy since I don't even remember where I read this, and don't really understand the topic, but I thought it was cool. Basically, it said on a spin manifold, a manifold where can do a lift from, for example an SO(3) twisted bundle to Spin(3), the weights of...

im buying books to get better at proofs so that i can tackle rudins analysis text. my question is, do i read spivak manifolds before or after rudin? a lot of sources list manifolds as a second year text, suggesting it is required reading. but some people also say it should be read after or...

I would like to make visualisations of calabi-yau manifolds, like this http://en.wikipedia.org/wiki/Calabi-Yau_manifold" (the image on the right). It would appear that http://www.povray.org/" is the appropriate tool (I suspect, after much Googling, that the image was created with POVRay)...

I would like to discuss this chapter with someone who has read the book. From looking at other books, I realize that Spivak does things a little differently. He seems to be putting less structure on his chains (for instance, no mention of orientation, no 1-1 requirement and so on), and as a...

Problem: given compact set C and open set U with C \subsetU, show there is a compact set D \subset U with C \subset interior of D. My thinking: Since C is compact it is closed, and U-C is open. Since U is an open cover of C there is a finite collection D of finite open subsets of U that...

Working through Spivak "Calculus on Manifolds." On p. 7, he explains that "the interior of any set A is open, and the same is true for the exterior of A, which is, in fact, the interior of R\overline{}n-A." Later, he says "Clearly no finite number of the open sets in O wil cover R or, for...

In the first problem set of chapter 1, problem 1-8(b) deals with angle preserving transformations. In the newest edition of the book the problem is stated If there is a basis x_1, x_2, ..., x_n and numbers a_1, a_2, ..., a_n such that Tx_i = a_i x_i, then the transformation T is angle...

Does anyone know of any research programs out there that are considering physical structure formation in terms of emergent, self-organizing statistical manifolds, and does so without starting from some reasonable first principles without any structure or preconception of manifold, or ad hoc...

In the attached pdf file i have a few questions on manifolds, I hope you can be of aid. I need help on question 1,2,6,7. here's what I think of them: 1. a) the definition of a smooth manifold is that for every point in M we may find a neighbourhood W in R^k of it which the intersection W...

Curves are functions from an interval of the real numbers to a differentiable manifold. Given a metric on the manifold, arc length is a property of the image of the curves, not of the curves itself. In other word, it is independent of the parametrization of the curve. In the case of the...

Can you define a space that is locally homeomorphic to an infinite dimensional Hilbert space analogously to how you define a (smooth) manifold by an atlas defining local homeomorphisms to R^n? So the charts would just map to the Hilbert space rather than R^n. Could the rest of the definition...

string theorists, there are two approaches to reducing 11 dimensions to 4, they are large and we are stuck on one, or they are compactified, too small to see. Which approach makes the most contact with physics? Is it possible to have 1-2 large dimensions and a 4-folded Yau-Calbi space...

Assume you have two manifolds M and N diffeomorphic to another. Also, there is a real-valued function f defined on M. What happens with f when you go from M to N? How is f related to N? thanks

Homework Statement Suppose that for every smooth Riemannian metric on a manifold M, M is complete. Show that M is compact. 2. The attempt at a solution I'm honestly not too sure how to start this question. If we could show that the manifold is totally bounded we would be done, but I'm...

Hi there, I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is...

Homework Statement If A\subset\mathbb{R}^{n} is a rectangle show tath C\subset A is Jordan-measurable iff \forall\epsilon>0,\, \exists P (with P a partition of A) such that \sum_{S\in S_{1}}v(S)-\sum_{S\in S_{2}}v(S)<\epsilon for S_{1} the collection of all subrectangles S induced by P such...