Manifolds Definition and 290 Threads

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...

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 what I've...

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...

Hi, everyone: I am doing some reading on the Frolicher Spec Seq. and I am trying to understand better the Kahler mflds. Specifically: What is meant by the fact that the complex structure, symplectic structure and Riemannian structure (from being a C^oo mfld.) are "mutually...

Could someone please help me with: if N,M are diffeomorphic manifolds, what property do they share that non-diffeomorphic manifolds do not share?. I have thought that if A,B were non-diffeomorphic (with dimA=dimB=n), certain functions (i.e, with their respective coord...

I am trying to understand differentiable manifolds and have some questions about this topic: We can think of a circle as a 1-dim manifold and make it into a differentiable manifold by defining a suitable atlas. For example two open sets and stereographic projection etc. would be the...

Please forgive any stupid mistakes I've made. On p.85, 4-5: If c: [0,1] \rightarrow (R^n)^n is continuous and each (c^1(t),c^2(t),...,c^n(t)) is a basis for R^n , prove that |c^1(0),...,c^n(0)| = |c^1(1),...,c^n(1)| . Maybe I'm missing something obvious, but doesn't c(t) =...

Why are Lie groups also manifolds?

What Lie groups are also Riemann manifolds? thanks

Let q and q' be sufficiently close points on C^oo manifold M. Then is it true that any C^oo curve c:[a,b]-->M where c(a)=q, c(q)=q' can be represented as c(t)=exp_{q}(u(t)v(t)) where u:[a,b]-->R,v:[a,b]-->TM_{q} and ||v||=1? My question comes from Chapter 9 corollary 16 and 17 of Spivak vol1...

what exactly are manifolds? I looked on wikipedia and I am getting the sense that its like n dimensional surface if that makes any sense.

Homework Statement I am trying to show that \vec{e'}_a = \frac{\partial x^b}{\partial x'^a} \vec{e}_b where the e's are bases on a manifold and the primes mean a change of coordinates I can get that \frac{\partial x^a}{ \partial x'^b} dx'^b \vec{e}_a = dx'^a \vec{e'}_a from the invariance...

http://www.sissa.it/fm/ncg07.html Workshop on Noncommutative Manifolds II Trieste NCG07 October 22-26, 2007 The Department of Mathematics of the University of Trieste and the International School for Advanced Studies (SISSA) organize a workshop on Noncommutative Manifolds. The workshop will...

A={ {{cos x, -sin x},{sin x, cos x}}|x \inR}, show that set A is smooth manifold in space of 2x2 real matrix. What is tangent space in unity matrix? My questions about problem: 1. What is topology here? (Because I need topology to show that this is manifold) 2. In solution they say that...

Is there some solved problem book about manifolds? (or where can I find solved problems on manifolds)

im thinking of taking in 2008 the second semester a course in analysis of manifolds. now some of the preliminaries although not obligatory, are differnetial geometry and topology, i will not have them at that time, so i think to learn it by my own, will baby rudin and adult rudin books will...

Homework Statement In Calculus on Manifold pp.83-84, Spivak writes that "if v_1,...,v_{n-1} are vectors in R^n and f:R^n-->R is defined by f(w)=det(v_1,...,v_{n-1},w), then f is an alternating 1-tensor on R^n; therefore there is a unique z in R^n such that <w,z>=f(w) (and this z is denoted v_1...

I need to following subjects about GRASSMANN MANIFOLDS,what do I? 1)introduction(together with details) 2)charts,atlas(together with details) 3)depended subjects

Fact: Spacetime is a curved pseudo-Riemannian manifold with a metric of signature (-+++). Fact: A manifold is a set together with a topology that is locally homeomorphic to R^n. Question: In the case of space-time, what is the set, what is the topology and what is n?

Hi, I have a question. Consider a differentiable manifold. This structure is imposed by requiring differentiability of the transition functions between charts of the atlas. Does requiring on top of that, linearity or affinity of the transition functions, result in any specific extra...

Let M be a diff. manifold, X a complete vectorfield on M generating the 1-parameter group of diffeomorphisms \phi_t. If I now define the Lie Derivative of a real-valued function f on M by \mathscr{L}_Xf=\lim_{t\rightarrow 0}\left(\frac{\phi_t^*f-f}{t}\right)=\frac{d}{dt}\phi_{t}^{*}f |_{t=0}...

My Solutions to "Tensors and Manifolds" Textbook Right now I am reading my current favourite book "Tensors and Manifolds with Applications to Relativity" by Wasserman, 1992. I am doing the exercises and typing out my solutions. I would like to share my solutions (with the questions typed out)...

According to my notes on SUSY 'as everyone knows, every continuous group defines a manifold', via \Lambda : G \to \mathcal{M}_{G} \{ g = e^{i\alpha_{a}T^{a}} \} \to \{ \alpha_{a} \} It gives the examples of U(1) having the manifold \mathcal{M}_{U(1)} = S^{1} and SU(2) has...

Does anyone know if there's worked out solution to the problems in spivak's calculus on manifolds? It's awfully easy to get stuck in the problems and for some of them I don't even know where to start... Also, if there isn't any, any good problem and 'SOLUTION' source for analysis on manifolds...

I know that the Riemann tensor vanishes in a flat space. And no amount of co-ordinate transformations can go from a flat space to a curved space. Does that mean there is no transformation that will go from, say Cartesian 2D, to (\theta,\phi), the co-ordinates usually used for the unit 2-sphere...

A manifold is a topological space which locally looks like R^n. Calculus on a manifold is assured by the existence of smooth coordinate system. A manifold may carry a further structure if it is endowed with a metric tensor. Why further structure? If have sphere or a cylinder I can...

(1) If A is an n x n matrix, then prove that (A^T)A (i.e., A transpose multiplied by A) is symmetric. (2) Let S be the set of symmetric n x n matrices. Prove that S is a subspace of M, the set of all n x n matrices. (3) What is the dimension of S? (4) Let the function f : M-->S be defined by...

Hi. I'm a bit stuck with that next question (and that's quite an understatement): Let f:M->N be a continuous map, with M and N smooth manifolds of dimensions m,n correspondingly. Define f*:C(N)->C(M) by f*(g)=g o f. Assume now that f*(C^infty(N)) subset C^infty(M). Then f is...

so i checked out Spivak's calculus on manifolds today, to work on while I'm in colorado this summer. i just finished up this semester with calc3 (multivariable), and I've take matrix theory and linear algebra as well. should I be good to go on this book at this point? I'd like to know since...

1) General question : Let's take a usual line : it's a 1D manifold in 2D space. The line is closed if there are no border points. (circle, aso...) Let suppose a usual surface : it's a 2D manifold enbedded in 3D space. The surface is closed if there are no border line. (sphere, torus...

So the equations of QM give eigenfunctions and eigenvalues. The eigenfunctions form a complete set with which any state is a combination of such. When measuring, the superposition of states collapse to one of the eigenfunctions. And the probability that some state with be measured in a...

hi, i encountered this term in julian barbour webpage and i will like it if someone can tell me more about them?

I want to know how many Calabi-Yau manifolds there are in each of the 5 superstring theories. Can you point me in the right direction?

i am trying to solve this problem: Give the paraboloid y_{3}=(y_{1})^2+(y_{2})^2 the structure of a smooth manifold. But i am unsure what it means by structure. Can anyone give me some help here?

I'm trying to understand the manifold properties of world-sheets in string theory. I'm told that world sheets are manifolds and that manifolds are locally Euclidean. So I would like to know the characteristics between the space-time coordinates of the world-sheet given as x&mu; verses the 2D...

what are they?