Manifold Definition and 336 Threads

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?

Homework Statement Establish the theorem that any 2-dimensional Riemann manifold is conformally flat in the case of a metric of signature 0. Hint: Use null curves as coordinate curves, that is, change to new coordinate curves \lambda = \lambda(x0, x1), \nu = \nu(x0, x1) satisfying...

Hey everybody! Physicists have no problem differentiating a function of many variables - in flat space R^n. But I don't like how many books don't give examples of how this done in a manifold- even if it may be easy when one finally understands it. For example, how do I differentiate a...

Hi, I have two questions: how can we prove a closed ball in R^n is manifold with boundry only using the definition being manifold with boundry. Also i want to ask C^/inf(M) is infinite dimensional where M is smooth manifold of dimension n>0.

Arguably, this is pure mathematical question, but most discussions of semi-riemannian manifolds are in the context of physics, so I post here. Can anyone state or point me to references discussing best known answers to the following: Given an arbitrary Semi-Riemannian 4-manifold, and an...

Suppose I have a highdimensional space \mathbb{R}^N that is sparesely populated by a finite set of samples \{ \mathbf{x} \}_{1 \le i \le k} , for example N = 500, k = 100. I assume the points x to be sampled from a n-manifold embedded in \mathbb{R}^N, where n << N. From a mathematical point of...

Hi, I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help! Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for...

I have a question about the definition of a manifold given in my analysis book. Here is the definition: Let 0 < k \le n . A k-manifold in \mathbb{R}^n of class C^r is a set M \subset \mathbb{R}^n having the following property: For each p in M, there is an open set V of M containing p...

Hi all! I am reading a book on Classical Electrodynamics (Hehl and Obukhov, Foundations of Classical Electrodynamics, Birkhauser, 2003). In this manual I found the following statement: ============ Consider a 4-dimensional differentiable manifold which is: -connected (every 2 points are...

I'm new to manifolds, so please forgive me if this sounds ignorant. I was just wondering whether the charts of a smooth manifold (within some atlas) always "overlap". If I'm not mistaken they map to open subsets of R^n, and being homeomorphisms should have the inverse image as open. But I'm not...

We all are familiar with the kind of differential geometry where some affine connection always exists to relate various tangent spaces distributed over the manifold, and from this connection two fundamental tensors, namely the Cartan's torsion and the Riemann-Christoffel curvature, arise. Is it...

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.

When we integrate a scalar map over a manifold M, what exactly are we measuring? If M is the unit circle in R^2, then regular Riemann integration of the function f = 1 over it will yield the volume of a cylinder of height 1. Okay, no problem. Now, if we integrate f = 1 over the unit circle in...

Given a manifold as algebraic variety, say sphere, how do we obtain possible metrics? how do we classify them? If spcaetime manifold is n-sphere, Einstein's vacuum (for now) equation would be some special metric among many other possible metrics? i'm curious what role Einstein's equation...

What is the least number of charts needed to specify a given smooth manifold, for simplicity of dimension 2? For instance the minimum number of charts to cover a torus, or a 2-sphere or a 2D 1-sheet hyperboloid? I would think it goes with the definition of manifold that in any case you need at...

I have a question regarding the following definition of convergence on manifold: Let M be a manifold with atlas A. A sequence of points \{x_i \in M\} converges to x\in M if there exists a chart (U_i,\phi_i) with an integer N such that x\in U_i and for all k>N,x_i\in U_i \phi_i(x_k)_{k>N}...

obviously in one coordinate neighborhood it can't.. I'm thinking of a line which smoothly develops into a surface : -----<< what particular properties would this object have.. Thanks :)

I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it? for example, i know...

I have the result that any 1 dim topological manifold is either R or S1. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices. i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the...

Homework Statement Let M be a differentiable manifold and g: \Re \times M \rightarrow M, (t,x) \rightleftharpoons g^{t}x be a map such that the following conditions are satisfied. i) g is a differentiable map. ii) The map \Re \rightarrow Aut(M), t \rightleftharpoons g^{t} is a...

Hi everyone, On the Wikipedia page for Tangent space there is a definition of the tangent space at a point x using equivalence classes of curves. It mentions that the tangent space TxM is in bijective correspondence with Rn. My first question is simply: is there an easy geometric way using...

I’m studying GR and am curious about manifold, atlas and charts. I have an idea for building a simple example, in one dimension, and wanted to ask if what I’m doing below is “legal”/correct. Imagine a space flight that can be divided into three segments: A-B: velocity starts at zero and...

Hello, I am learning about smooth manifolds through Lee's text. One thing that I have been pondering is describing manifolds such as |x| which are extremely well behaved but not smooth in an ordinary setting. It is simple to put a smooth structure on this manifold, however that is...

Is it possible that Lorentz invariance is just a lower limit of a larger manifold that has a priveleged frame? Even if Bell's experiments can't transmit signal faster than light. The spirit of relativity is still violated by say instantaneous correlation between 10 billion light years. As...

Can anyone help with finding the length of a circle (theta) =pi/2 (latitude 90') on the unit sphere. I know it is related to the equation L= integral from 0 to T of Sqrt(g_ij (c't,c't)) The formula is on the wikipedia page called Riemannian manifold so you can get a better idea what it...

Can someone please explain to me how this formula integral from 0 to T of sqrt(g_ij c'(t) c'(t)) I have seen it on wikipedia but don't know how to actually implement the formula.

Homework Statement Prove the triangle inequality theorem \leftX-Y\right\rfloor\lfloor\leq\leftZ-Y\right\rfloor\lfloor+\leftY-X\right\rfloor\lfloor. (my computer just shows the latex as a bunch of script so i don't know if that came out right. Homework Equations The Attempt at a...

hello friends :smile: I have a question about the compactness of the tangent bundle: assume that the manifold M is compact, does it make necessarily TM compact ? if not TM, a submanifold of TM (precisely a submanifold of vector norm equal to 1) can be compact?

Homework Statement In which points the surface \{\left(x,y,z\right)\in\Re^{3}|x^{3}-y^{3}+xyz-xy=0\right\} is a differentiable manifold (subvariedad diferenciable in spanish). Calculate its tangent space in the point (1,1,1). Homework Equations NA The Attempt at a Solution I've...

Homework Statement As the title suggests, I need to show that every manifold is regular. There's probably something wrong with my proof, since I didn't use the Hausdorff condition, and the book almost explicitly states to do so. The Attempt at a Solution So, a m-manifold is a...

hello friends my question is: if we have M a compact manifold, do we have there necessarily TM compact ? thnx .

Mass can curve space-time. Is it possible that space-time around a black hole is so badly curved that it forms a closed 4D manifold?

Homework Statement Im taking a dynamics course and I am using The strogatz book Non-linear Dynamics and Chaos I need to solve a problem that is similar to problem 6.1.14 Basically it consist in the following You have a saddle node at (Ts,Zs) which is (1,1). Consider curves passing through...

Is there any non-orientable one-dimensional manifold ? If not, how to prove it? Thanks!

let M be a manifold and g a metric over M . is it true that every subbundle from M must have the same metric g ?

Okey, I have problem with the foundation of lie algebra. This is my understanding: We have a lie group which is a differentiable manifold. This lie group can for example be SO(2), etc. Then we have the Lie algebra which is a vectorspace with the lie bracket defined on it: [. , .]. This...

Hi, I am using a model that estimates pressure (P) in an intake manifold. I think there is a mistake in its equations but I cannot find it. To simplify the problem we can make the following assumptions: - Only air fills the manifold: air comes into the manifold through the throttle (mass...

Homework Statement Let's assume that M is a compact n-dimensional manifold, then from Whitney's Immersion Theorem, we know that there's an immersion, f: M -> R_2n, and let's define f*: TM --> R_2n such that f* sends (p, v) to df_x (v). Since f is an immersion, it's clear that f* must be...

Background. We define vectors in general relativity as the differential operators \frac{\cdot}{d\lambda}=\frac{dx^\mu}{d\lambda}\frac{\cdot}{\partial x^\mu} which act on infinitessimals--dual vectors, df=\frac{\partial f}{\partial x^\mu} dx^\mu \ , as linear maps to reals. However, both...

First let me write out the definition of a manifold given in my book: Let k > 0 . A k-manifold in \mathbb{R}^n of class C^r is a subspace M of \mathbb{R}^n having the following property: For each p \in M , there is an open set V \subset M containing p , a set U that is open in...

Suppose M and N are smooth manifold with M connected, and F:M->N is a smooth map and its pushforward is zero map for each p in M. Show that F is a constant map. I just remember from topology, the only continuous functions from connected space to {0,1} are constant functions. With this be...

Most definitions I've seen for a manifold are based on the idea that small neighborhoods are homeomorphic to \mathbb{R}^n. To me this feels a little like defining a bicycle as a car that's missing the engine and both the wheels on one side. The real number system is this big, sophisticated piece...

In Lee's Intro to topological manifolds, p.105, it is written that every manifold of dimension 3 or below is triangulable. But in dimension 4, threre are known examples of non triangulable manifolds. In dimensions greater than four, the answer is unknown. But in Bott-Tu p.190, it is written...

If k is an integer between 0 and min(m,n),show that the set of mxn matrices whose rank is at least k is an open submanifold of M(mxn, R).Show that this is not true if "at least k"is replaced by "equal to k." For this problem, I don't understand why the statement is not true if we replace "at...

For a game I am thinking about making I would need to know how to project points from a differentiable bounded 3-manifold to a Euclidean plane (the computer screen). The manifold would be made from a 3-dimensional space with two balls cut out of it and a hypercylinder glued onto it at the holes...

In a different thread, JesseM raised ( https://www.physicsforums.com/showpost.php?p=2858281&postcount=37 ) what I thought was an interesting question: can a manifold (over the real numbers) contain points that are infinitely far apart? Since a bare manifold doesn't come equipped with a metric...