Manifolds Definition and 282 Threads

My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface? Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...

Hello, let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$ be two submanifolds. We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation $$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...

This is a refinement of a previous thread (here). I hope I am following correct protocol. Homework Statement I am studying Spacetime and Geometry : An Introduction to General Relativity by Sean M Carroll and have a question about commutators of vector fields. A vector field on a manifold can...

Homework Statement I am studying Spacetime and Geometry : An Introduction to General Relativity by Sean M Carroll and have a question about commutators of vector fields. A vector field on a manifold can be thought of as differential operator which transforms smooth functions to smooth functions...

I am studying Analysis on Manifolds by Munkres. He introduces improper/extended integrals over open set the following way: Let A be an open set in R^n; let f : A -> R be a continuous function. If f is non-negative on A, we define the (extended) integral of f over A, as the supremum of all the...

Hello. I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20. It states that: Let A be an n by n matrix. Let h:R^n->R^n be the linear transformation h(x)=A x. Let S be a rectifiable set (the boundary of S BdS has measure 0) in R^n. Then v(h(S))=|detA|v(S)...

I am reading "Road to Reality" by Rogen Penrose. In chapter 15, Fibre and Gauge Connection ,while going through Clifford Bundle, he says: .""""...Of course, this in itself does not tell us why the Clifford bundle has no continuous cross-sections. To understand this it will be helpful to look at...

The Reshetikhin-Turavev construction comes with an invariant that is sometimes called the Reshetikhin-Turaev Invariant. I'm currently attempting to wrap my head around this construction but was hoping for a sneak peak to help motivate me. My question is, what does the Reshetikhin-Turaev...

I have been stuck several days with the following problem. Suppose M and N are smooth manifolds, U an open subset of M , and F: U → N is smooth. Show that there exists a neighborhood V of any p in U, V contained in U, such that F can be extended to a smooth mapping F*: M → N with...

So in string theory at each point of Minkowski spacetime we might have a 3 dimensional compact complex Calabi–Yau manifold? We can have curved compact spaces without complex numbers I assume, what is interesting or special about complex compact spaces? Thanks!

Hello! I was thinking about the Riemann curvature tensor(and the torsion tensor) and the way they are defined and it seems to me that they just need a connection(not Levi-Civita) to be defined. They don't need a metric. So, in reality, we can talk about the Riemann curvature tensor of smooth...

Hello! Since connections in general do not require that we have a Riemannian manifold, but only a smooth manifold, I find it kind of weird that the only examples of connections that I find in the internet are those which use the Levi-Civita connection. So, I wanted to know of any examples of...

Suppose ##p: C-->X## is a covering map. a) If ##C## is an n-manifold and ##X## is Hausdorff, show that ##X## is an n-manifold. b) If ##X## is an n-manifold, show that ##C##is an n-manifold c) suppose that ##X## is a compact manifold. Show that ##C## is compact if and only if p is a finite...

Let ##M## and ##N## be connected n-manifolds, n>2. Prove that the fundamental group of ##M#N## (the connected sum of ##M## and ##N##) is isomorphic to ##\pi(M)* \pi(N)## (the free group of the fundamental groups of ##M## and ##N##) This is not for homework, I was hoping to get some insight...

I am applying a Green's probabilistic elastodynamic tensor with relativistic manifold extensions to solve a pull out of a smoothly shaped deformable spheroid from a stiff inhomogenous deformable quasi-brittle host. This involves a Hooke's law tensor, a relativistic manifold Ricci tensor, a...

Hi I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...

Any open subset of ##\mathbb{R}^{n}##; The n-Sphere, ##\mathbb{S}^n##; The Klein Bottle. I guess they don't have a boundary, as a neighborhood of any point of them is homeomorphic to ##\mathbb{R}^n##. I'd like to know whether my guess is correct and whether the reason I'm giving for them not to...

Why manifolds in General (and Special) Relativity have to be open? Would this be because an open manifold have a continuous interval? (i.e. an interval with no interruptions)

Are the metrics for say the Calabi-Yau manifolds of string theory, assuming they have a metric, dynamic in the sense that a vibrating string interacts with the compact space causing the metric to change where there is a string, even if only a tiny amount? Thanks!

In my ignorance, when first learning, I just assumed that one pushed a vector forward to where a form lived and then they ate each other. And I assumed one pulled a form back to where a vector lived (for the same reason). But I see now this is idiotic: for one does the pullback and pushforward...

Manifolds of contant curvature are conformally flat. I'm trying to find a stronger claim related to this for manifolds of dimension >2. Does anyone knows if for instance Riemannian manifolds(of dimension >2) with non-constant curvature are necessarily not conformally flat, or maybe something...

Hello every one . first of all consider the 2-dim. topological manifold case My Question : is there any difference between $$f \times g : R \times R \to R \times R$$ $$(x,y) \to (f(x),g(y))$$ and $$F : R^2 \to R^2$$ $$(x,y) \to (f(x,y),g(x,y))$$ Consider two topological...

So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...

I'm interested in this subject. This is a graduate text and I believe the prereqs are mostly a math degree, which I somewhat have(B.S in Applied Math from a few years back). The thing is, I forgot details about things. For example, I know how to do an epsilon delta proof and can read one when...

1. The problem statement, all variables and given Before I try to work through the book, it would be great to have a list of typos, if there are any. Homework EquationsThe Attempt at a Solution

I would like to know at what level is the book Tensors and Manifolds by Wasserman is pitched and what are the prerequisites of this book? Given the prerequisites, at what level should it be (please give examples of books)? If anyone has used this book can you please kindly give your comments and...

I have a question regarding the usage of notation on problem 2-11. Find ##f'(x, y)## where ## f(x,y) = \int ^{x + y} _{a} g = [h \circ (\pi _1 + \pi _2 )] (x, y)## where ##h = \int ^t _a g## and ##g : R \rightarrow R## Since no differential is given, what exactly are we integrating with...

While Minkowski space and Euclidean space both have identically zero curvature tensors it seems that a flat Lorentz manifold in general, may not admit a flat Riemannian metric. Such a manifold is the quotient of Minkowski space by the action of a properly discontinuous group of Lorentz...

In my introduction to manifolds the following is stated: Polar coordinates (r, phi) cover the coordinate neighborhood (r > 0, 0 < phi < 2pi); one needs at least two such coordinate neighborhoods to cover R2. I do not understand why two are needed. Any point in R2 can be described by polar...

I am really struggling with one concept in my study of differential geometry where there seems to be a conflict among different textbooks. To set up the question, let M be a manifold and let (U, φ) be a chart. Now suppose we have a curve γ:(-ε,+ε) → M such that γ(t)=0 at a ∈ M. Suppose further...

As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...

I'm fairly new to differential geometry (learning with a view to understanding general relativity at a deeper level) and hoping I can clear up some questions I have about coordinate charts on manifolds. Is the reason why one can't construct global coordinate charts on manifolds in general...

I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the...

I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation). Later, I run into the...

No question this time. Just a simple THANK YOU For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups. My math background was very deficient: I am a 55 year old retired (a good life) professor of...

I have found the following entry on his blog by Terence Tao about embeddings of compact manifolds into Euclidean space (Whitney, Nash). It contains the theorems and (sketches of) proofs. Since it is rather short some of you might be interested in.

I have found this paper on the internet and think it might be interesting for some on this forum because there are frequently questions similar to the ones the paper tries to answer. http://arxiv.org/abs/1605.00890 http://arxiv.org/pdf/1605.00890v1.pdf

Hello, As you might discern from previous posts, I have been teaching myself: Calculus on manifolds Differential forms Lie Algebra, Group Push forward, pull back. I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...

This is from Jackson, "Electrodynamics" a field is a fuction mapping phi: M -> T, x -> phi(x) from a base manifold M into a target manifold T. field X: R3 * R1 -> R3 x(r,t) ->X(x)I think this is eucledian R4 to R3 so I wonder why Jackson explained this with the concept of manifolds? Is it...

Hello! Good morning to all forum members! I am studying general relativity through the wonderful book: "General Relativity: An Introduction for Physicists" by M.P. Hobson (Cambridge University Press) (2006). My question is about Riemannian manifolds and local cartesian coordinates (Chapter 02 -...

I am reading John M. Lee's book: Introduction to Smooth Manifolds ... I am focused on Chapter 1: Smooth Manifolds ... I need some help in fully understanding Example 1.3: Projective Spaces ... ... Example 1.3 reads as follows:My questions are as follows:Question 1In the above example, we...

I am reading "An Introduction to Differential Topology" by Dennis Barden and Charles Thomas ... I am focussed on Chapter 1: Differential Manifolds and Differentiable Maps ... I need some help and clarification on an apparently simple notational issue regarding the definition of a chart...

As many of you know, using the stereographic projection one can construct a homeomorphism between the the complex plane ℂ1 and the unit sphere S2∈ℝ3. But the stereographic projection can be extended to the n-sphere/n-dimensional Euclidean space ∀n≥1. Now what I am talking about is the the...

Does the words "Quantum Manifolds" make any sense? Can quantum occurs on a manifold? Or do you automatically equate manifolds to lorentzian manifold and it becomes a problem of quantum gravity? Or can there be manifolds not related to spacetime.. so can quantum on manifold make sense?

Hi, I am just about to finish working through the integration chapter of calculus on manifolds, and I am wondering whether it would be better to get spivaks first volume of differential geometry (or another book, recommendations?) and start on that, or to finish calculus on manifolds first...

Dear Physics Forum friends, I am currently trying to purchase Munkres' Analysis on Manifolds to replace the vector-calculus chapters of Rudin-PMA, which is quite unreadable compared to his excellent chapters 1-8. I know that there is a paperback-edition for Munkres, but I heard that the...

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...

I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the Ricci rotation coefficients...

Dear all We all agree that a manifold is globally non euclidean but locally it is. So we can find near each point a hemeomorphic to an open set of euclidean space of the same dimension as the manifold. This is a general definition for all manifold to follow. Then what is the difference between...

Hi How can a person chart a manifold if he does not know how the manifold looks like ? E.g. The 2-sphere manifold can have 2 charts and symmetric charts with the chart goes like this ( theta from zero to pi , psy from minus pi to pi ) but the problem for unknown manifold , e.g. In general...