Manifold Definition and 336 Threads

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...

Hello, (Continuing learner: new to all of this: patience requested, please.) What is the difference between a manifold and a submanifold? I have read and now been working with each. I understand the technical definitions (I think I do... but I do not have an intrinsic FEELING for these...

I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...

Why does higher intake manifold pressure results in increase in the degree of spark advancement? Similarly, why is the torque higher for higher intake manifold pressure?

Consider the following definition: (##M## denotes a manifold structure, ##U## are subsets of the manifold and ##\phi## the transition functions) Def: A smooth curve in ##M## is a map ##\gamma: I \rightarrow M,## where ##I \subset \mathbb{R}## is an open interval, such that for any chart...

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...

I'm just trying to understand something about manifolds. What is meant when a manifold doesn't have boundary? I thought the boundary was where the manifold "ends" so to speak. Like a boundary point, something where you take a small nhbd (neighborhood) and you get something inside the set and...

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...

I am relatively new to the concept of differential geometry and my approach is from a physics background (hoping to understand general relativity at a deeper level). I have read up on the notion of diffeomorphisms and I'm a little unsure on some of the concepts. Suppose that one has a...

I've been reading up on the definition of a tangent bundle, partially with an aim of gaining a deeper understanding of the formulation of Lagrangian mechanics, and there are a few things that I'm a little unclear about. From what I've read the tangent bundle is defined as the disjoint union of...

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of dimension $n-r$ in $\mathbf R^n$. We imitate the proof of Lemma 1 on pg 11 in Topology From A...

Hello : let be a differential manifold C^{\infty} : M of dimension n. I choose a point p. In this point I can defined the tangent space. It's a vectoirial space of dimension n, I'll talk about it in a precedent thread, . This space is in bijection with the derivation space : each derivation...

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator [itex] X[\itex] acting on some function [itex]f:M\rightarrow\mathbb{R}[\itex] at a point [itex]p\in M[\itex] (where [itex]M[\itex] is an...

I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a function f:M\rightarrow N is independent of the coordinate chart that we use". He shows this is...

I'm trying to understand what exactly it means by some tensor field to be 'well-defined' on a manifold. I'm looking at some informal definition of a manifold taken to be composed of open sets ##U_{i}##, and each patch has different coordinates. The text I'm looking at then talks about how in...

Hi, this is just a review exercise. Let M,N be n- and m- manifolds respectfully , so that the product manifold MxN is orientable. I want to show that both M,N are orientable. I could do some computations with product open sets of ##\mathbb R^n ## , or work with orientation double-covers...

Hi all, Could anyone please clarify something for me. PCA of a data matrix X results in a lower dimensional representation Y through a linear projection to the lower dimensional domain, i.e Y=PX. Where rows of P are the eigenvectors of X. From a pure terminology point of view is it correct...

Consider a flat Robertson-Walker metric. When we say that there is a singularity at $$t=0$$ Clearly it is a coordinate dependent statement. So it is a "candidate" singularity. In principle there is "another coordinate system" in which the corresponding metric has no singularity as we...

Hi All, I am trying to figure out the details on giving a surface S a hyperbolic metric with geodesic boundary, i.e., a metric of constant sectional curvature -1 so that the (manifold) boundary components, i.e., a collection of disjoint simple-closed curves are geodesics under this metric. So...

In a Google image search image search for "Ricci Flat manifolds" I came up with Calabi-Yau manifolds for dummies at, http://universe-review.ca/R15-26-CalabiYau01.htm Lots of pictures and important terms. Other good stuff there as well, click on home button.

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!

Is it true that the atlas for a torus can consist of a single map while the atlas for a sphere requires at least two maps? Can we ever get by with a single map for some Calabi–Yau manifolds assuming that question makes sense? If not is there some maximum number required? Thanks for any help!

Does it make sense to ask if we can "comb the hair" on a Calabi yau manifold? Hope this is the right place for this question. Thanks for any help!

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)...

Let n <= m and G:=Gr(n,m) be the (real) Grassmanian manifold. I understand the topology of the simplest case, that of projective space, and am wondering if there is a way to interpret the topology of the G to similar to projective space, with the according generalizations needed. If V^n is an...

Some words before the question. For two smooth manifolds M and P It is true that T(M\times P)\simeq TM\times TP If I have local coordinates \lambda on M and q on P then (\lambda, q) are local coordinates on M\times P (right?). This means that in these local coordinates the tanget vectors are...

Homework Statement I am confused if there is any standard way to check what should be the line element $$ds^{2}$$ when the dimensions are more than three ( since we don't have the option to draw things as we usually do in case of 3d or less dimensional cases. I am following Hobson's book. I am...

Hello, I want to have a basic understanding. When we speak 10 dimension in String Theory, do we mention: (a) The six dimension of Calabu Yau manifold and (4) the four dimension space-time?

Can somebody explain to me what is a manifold.Also what it means for a space to be curved and how we define curvature.I know that a sphere is a curved 2d object, can a curved 3d object live in 3-dimensional space?

Hi, I have an exercise whose solution seems too simple; please double-check my work: We have a product manifold MxN, and want to show that if w is a k-form in M and w' is a k-form in N, then ##(w \bigoplus w')(X,Y)## , for vector fields X,Y in M,N respectively, is a k-form in MxN. I am...

While reading about sheaves, I came across a beautiful definition of a manifold. An ##n##-manifold is simply a locally ringed space which is locally isomorphic to a subset of ##(\mathbb{R}^n, C^0)##. However, I don't see how this guarantees a manifold to be Hausdorff. Would someone please...

Is it possible for a riemann manifold to change its curvature? In practice could the universe in general change its curvature by time? (let's say in the past it was negative and today it's almost flat tending to positive); If not which theorem disproves it?

On the spacetime manifold in general relativity, one chooses a basis at a point and express it by the partial derivatives with respect to the four coordinates in the coordinate system. And then the basis vectors in the dual space will be the differentials of the coordinates. Why do one do that...

I k-foliation of a ##n##-manifold ##M## is a collection of disjoint, non-empty, submanifolds who's union is ##M##, such that we can find a chart ##(U,x^1, \ldots mx^k, y^{k+1}, \ldots, y^n)=(\phi, (x^\mu, y^\nu))## about any point with the property that setting the ##n-k## last coordinates equal...

"2D slice of a 6D Calabi-Yau manifold", and other? Mathematically what does it mean to take a "2D slice of a 6D Calabi-Yau manifold"? Part of quote taken from the top of, http://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold Is there a finite number of slices of a 6D Calabi-Yau manifold...

According to Isham (Differential Geometry for Physics) at page 115 he claims: "If X is a complete vector field then V can always be chosen to be the entire manifold M" where V is an open subset of a manifold M. He leaves this claim unproved. A complete vector field is a vector field which...

Hi friends, I was wondering about the following - in GR texts we always see these penrose diagrams and some line representing the horizon and all these timelike , spacelike curves and all that ... but the picture that I have of GR is just that of a smooth 4 manifold endowed with a metric . Can...

So the question is simple: is GL_n(\mathbb C) the biggest complex manifold in GL_{2n}(\mathbb R)? Or is there another one that strictly contains it?

I've been trying to prove that the closed unit ball is a manifold with boudnary, using the stereographic projection but I cannot seem to be able to get any progress. Can anyone give me a hint on how to prove it? Thanks in advance :)

Hi I am working on a robot that has a spinning 3D laser scanner. It rotates about two axis and collects data. In one axis it has full 3D rotation and in another axis it has limit rotation. Now the read world points collect by this laser scanner is not unifomaly distributed but if...

Left invariant fields on a group G satisfies a lie algebra; say we have an n-dimensional Lie algebra for which the fields ##{X_1, \ldots , X_n}## is a basis. Let these satisfy the algebra ##[X_a, X_b] = c_{ab}^c X_c##. Suppose now that we have a Riemannian manifold with killing vectors...

I've been away from the forum for a while working on an interesting project developing an open source visualization system for spatial manifolds that have four dimensions. I have two primary lines of questioning that stem from this work. 1) I know that Gerris is an open source solution for...

In the definition of smooth manifolds we require that the transition functions between different charts be infinitely differentiable (a circle is an example of such a manifold). Topological manifolds, however, does not require transitions functions to be smooth (or rather no transition functions...

Suppose we have a pseudo-riemannian 4 manifold S (sometimes also called a Minkowskian manifold) that is without boundary, and not simply connected. Suppose there is at least one pseudo-riemannian 4 manifold M (also without boundary) that has S as a regular submanifold, preserving all geometry...

By the well-known Whitney embedding theorem, any manifold can be embedded in \mathbb R^n. You might have also heard the Nash embedding theorem, which basically says that this is still true for Riemannian manifolds (i.e. now we demand the metric is induced from \mathbb R^n). So fine, any...

Broad title, but really a specific question that I thought should be straightforward, but got stuck. Consider the geodesics of form t=contant, r>R, in exterior SC geometry in SC coordinates. These are spacelike geodesics. If we consider this geometry embedded in Kruskal geometry, it is easy to...

The usual definition of an n-dimensional topological manifold M is a topological space which is 'locally Euclidean', by which we mean that: (1) every point in M is contained in an open set which is homeomorphic to ##\mathbb{R}^n##. (2) M is second countable. (3) M is an Hausdorff space...

Hi, All: Say S is a submanifold of an ambient, oriented manifold M; M is embedded in some R^k; let ## w_m ## be an orientation form for M. I'm trying to see under what conditions I can orient S , by contracting ## w_m ## , i.e., by using the interior product with the "right" vector...

Hi so I was just wondering if the metric g=diag(-e^{iat},e^{ibx},e^{icy}) (where a,b,c are free parameters and t,x,y are coordinates) corresponds to a complex manifold (or is nonsensical), and what the manifold looks like?