Manifold Definition and 318 Threads

Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc...

Hello, I have troubles formulating this question properly. So I will explain it through one example. If we consider the Lie group R=SO(2) of rotations on the plane, we know that we can find a manifold on which the group SO(2) acts regularly: this manifold is the unit circle in ℝ2. In fact...

Hello, I read from several sources the statement that the set of points M\inℝ2 given by (t, \, |t|^2) is an example of differentiable manifold of class C1 but not C2. Is that true? To be honest, that statement does not convince me completely, because in order for M to be a manifold, we should...

Originally I asked on another thread whether the GR effects on light can be described as light passing through a medium of varying density- so exerting its effects by refraction. A.T. kindly posted the following links: http://www.newscientist.com/article/dn24289#.UlsOR-B3Zmh...

I want to be able to formulate x^{n} coordinate system. x^{n} =(x^{1}, x^{2}, x^{3}, x^{4}) How do you do this when the Riemann Manifold is not rectangular or spherical? Also how do you differentiate with respect to "s" in that case. \frac{dx^n}{ds}

Mod note: Posts split off from https://www.physicsforums.com/showthread.php?p=4468795 Hi, WN, might the OP be referring to GR instead of SR, more specifically to the expanding FRW universe in which it is impossible to even consider the notion of exansion without agreeing about an "everywhere...

I understand that SR and GR should not be dependent on the coordinate system. But does GR depend on the existence of a manifold? As I recall, GR is formulated with a metric tensor. And tensors are only defined on manifold. Or is there a manifold independent GR?

Consider a discrete set of ##k## points. First, is it a manifold? I know that a manifold is a topological space that contains a neighborhood homeomorphic to Euclidean space for each point. Can we just consider each point's neighborhood to be a set containing only that point? Second, would the...

Hi all! I was wondering if \partial_1\partial_2f=\partial_2\partial_1f in a Riemannian manifold (Schwartz's - or Clairaut's - theorem). Example: consider a metric given by the line element ds^2=-dt^2+\ell_1^2dx^2+\ell_2^2dy^2+\ell_3^2dz^2 can we assume that...

I'm asking because I think of Minkowski space as a manifold with a Riemannian metric. However, I've also seen treatments in which an event in spacetime is chosen as origin, and special relativity treated as a vector space given the choice of origin. Does this mean that a vector space is a...

Hello All, I am drawing up a design for a water-cooled exhaust manifold for a small engine. It would be a simple design of an exhaust exit pipe mounted to a flange, with a tubular copper heat exchanger liquid filled coil wrapped around the exit pipe right near the cylinder. The copper would...

Why the figure of eight is not a manifold? I have read somewhere that if we remove the crossing point than the the figure of eight becomes disconnected, but by removing one point in \mathbb{R}^2 it's still connected. Is there any other proof without removing the crossing point?

I have a few questions about manifolds. According to Wikipedia.org, A topological manifold is a locally Euclidean Hausdorff space. First question, does locally Euclidean mean that there are a continuous set of points in order that they can be mapped to an infinite set of coordinates in the...

Can anyone explain the concept of manifold and Reimanni manifold in plain language ?? And what are its applications?? Thanks

In general terms a manifold can be defined simply as a topological space locally resembling Euclidean space with the resemblance meaning homeomorphic to Euclidean space, plus a couple of point set axioms that avoid certain "patological" manifolds and that some authors reserve for the definition...

I am researching a hypothesis and looking for anyone who is familiar with differential topology (specifically Einstein manifolds). I have access to the Besse book Einstein Manifolds but am also looking for any open questions in differential topology that I am not aware of. I am attempting to...

I'm trying to work out the following problem: Find the first two terms of the power series expansion for the volume of a ball of radius r centered at p in a Riemannian Manifold, M with dimension n. We are given that Vol(B_r(p)) = \int_S \int_0^r \det(d(exp_p)_{tv})t^{n-1}\mathrm{d}t...

Hi, All: I'm trying to show that the Mapping torus of a manifold X is a manifold, and I'm trying to see what happens when X has a non-empty boundary B. Remember that the mapping torus M(h) of a space X by the map h is constructed like this: We start with a homeomorphism h:X-->X (we...

Let L_{\theta} be the line passing through the point z(\theta)=(\cos\theta,\sin\theta) on the unit circle at angle \theta and with slope \frac{1}{2}\theta. The mobius band is M={(z,v):z\in S^{1},v\in L_{\theta}} my question is , why M is a mobius band?

I am having trouble getting a set definition of what constitutes a manifold for example , I have the real plane R^2, and the sphere s = {(x,y,z)|(x,y,z)£R^2, x^2+y^2+z^2=1} Note £, is meant to be "element of". And I have a continuous function f mapping the real plane onto s such that...

Hi all, I am trying to understand geometric flows, and in particular the Ricci flow. I understand how to calculate the metric tensor from the parametrization of a surface, but I am facing a problems in the concept phase. A metric tensor's purpose is to provide a coordinate invariant...

When talking cosmology what exactly is a manifold, is there any observational evidence that we live in one ?

I have a small pump system for an aquarium, and have decided to build a small manifold. The total volume of the manifold and the tubing is 2.9828 cu inches. Here is my pump specs: Inflation time < 8s (from 0-200 mmHg in a 100cc tank) Air flow > 70mL/min Max pressure > 360mmHg Leakage max...

Would someone here be able to write down for me an example of a metric on a manifold with both macroscopic dimensions, and microscopic "curled up" dimensions with some radius R ? Number of dimensions and coordinates used don't matter. Not going anywhere with this, I am just curious as to how...

I've posted a bunch of analysis questions as of late. I'm going to change things up a little bit and ask something that involves manifold theory. Here's this week's problem: ----- Problem: (i) Let $\omega$ be a 1-form. Use the structure equations \[\begin{aligned}d\theta^1 &=...

I am not sure -- a manifold is locally connected and has countable basis? There is an Exercise in a book as following : Given a Manifold M , if N is a sub-manifold , an V is open set then V \cap N is a countable collection of connected open sets . I am asking why he put this exercise...

I can't seem to find on the web a site that gives the Z cohomology of the infinite dimensional Grassmann manifold of real unoriented k planes in Euclidean space. I am interested in computing the Bockstein exact sequence for the coefficient sequence, 0 -> Z ->Z ->Z/2Z -> 0 to see which...

Hi there, I find that the term 'manifold' appears in many book of statistical physics or classical mechanics while talking about phase space. I try to search the explanation on online but it is quite abstract and hard to understand what's manifold really refers to. Can anyway explain this a...

We know that a Clifford torus is parameterized in 4D euclidean space by: (x1,x2,x3,x4) = (Sin(theta1), Cos(theta1), Sin(theta2), Cos(theta2)) {0<=theta1 and theta2<2pi} Consider that a clifford torus is the immediate result of Circle * Circle Now, have you encountered a similar manifold...

Homework Statement If β=0 the neurone model is \dot{u}= -u \dot{v}= v2 + v - u + \delta If \delta = 1/4 it has critical point (0,-1/2) Transform the system so that the critical point is at the origin so let \bar{v} = v +1/2 and find the equations of motion for (u,\bar{v}) Homework...

The theorem supposedly states: A space cannot be bent unless it is a manifold which is embedded in a space of at least one higher dimension. Does anyone know if this theorem actually exists? If it does, I would appreciate a reference to its name and proof.

It's always been my understanding that given the existence of BH singularities and the initial BB singularity our universe couldn't be geodesically complete. But then one of the premises of our cosmology models is that the universe is homogeneous, and all homogeneous manifolds are geodesically...

i see the definition of differential manifolds in some book for example, NAKAHARA but what is the definition of manifold in general! and what the definition of topological manifold.

From the things I've read on manifold geometry,metric is a property of the manifold.Maybe you can call it intrinsic. But consider e.g. a euclidean manifold and two coordinate systems on it.say,cartesian and spherical.As you know,the metric for this two coordinate systems is different. So what?Is...

Difference between hilbert space,vector space and manifold?? Physically what do they mean? I m really confused imagining them..Explanation with example would help me to understand there application ..THanks in advance

Homework Statement Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space. Homework Equations A topological manifold is a Hausdorff space that are locally Euclidean, i.e., there's an n such that for each x, there's a neighborhood...

In SR the points in Minkowski space correspond to events. I recently read in a GR lecture note that the points on the manifold do NOT correspond to events like in SR (the author even says the points don't have a direct physical meaning). So what do they represent then? And if I continuously...

I'm just learning manifold geometry and tensor analysis.From the things I've understood till now,an idea came into my mind but I can find it or its negation no where.So I came to ask it here. I can't explain how I deduced this but I think there should be sth like a natural coordinate for a...

Is it true, that if A and B are oriented manifolds with boundary, having dimensions n and m respectivelly, then the boundary of A\times B is \partial(A\times B)=\partial A\times B + (-1)^n A\times \partial B? If not, then what can we say about the boundary of product manifolds? Could someone...

Here is the situation I am concerned with - Consider a smooth curve g:[0,1] \to M where M is a topological manifold (I'd be happy to assume M smooth/finite dimensional if that helps). Let Im(g) be the image of [0,1] under the map g . Give Im(g) the subspace topology induced by...

Hi, Why R mod 2pi is a Compact Manifold? Isn't this like a real line which is not compact? How should we prove it using a finite sub-cover for this manifold? bah

Just a introduction to my project, I have designed and building a 69 Volkwagen Bug with a Version 8 subaru Sti Engine, The pan is built with tube braces and coil over suspension, I built a tall back bone frame for strength and for the rest of the frame to be lower for ride height and suspension...

I'm looking at prop 19.5 of Taylor's PDE book. The theorem is: If M is a compact, connected, oriented manifold of dimension n, and a is an n-form, then a=dB where B is an n-1 form iff the ∫a over M is 0. I'm trying to understand why a=dB implies ∫a = 0. If M has no boundary, than this...

I am looking for some intuition into a way of looking at the Gauss curvature on a surface that describes it as the divergence of a potential function - at least locally. I am not sure exactly what the intuition is - but this way of looking at things seems suggestive. Any insight is welcome. In...

i want to model air flow in the inlet manifold using CFD ,how do i predict how flow variations and other parameters affect engine performance ? i know how it affects but how do i show it virtually

Hi all, To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric. Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics...

Hi, I want to show that the set of boundary points on a manifold with boundary is well defined, i.e the image of a point on a manifold with boundary can not be both the interior point and boundary point on upper half space. To do this, it is enough to show that R^n can be homeomorphic to...

I came across this problem in Willard, General Topology 18H.2, where it says "For compact n-manifold X, prove that X is a product of spheres" I have no idea how to rigorously do it... I searched internet for this, but has no hope... any helps?

Helmholtz resonator? Intake manifold design Hi guys, I'm doing a bit of research into engine intake manifolds. The information I've found so far looks staright forward, but it's given out by turners and cottage industry designers, rather than engineers or physicists, so it's basic. I...

how to extend a locally defined function to a smooth function on the whole manifold ,by using a bump function?