In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.
One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics and augmented-reality given the need to associate pictures (texture) to coordinates (e.g. CT scans).
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
The study of manifolds requires working knowledge of calculus and topology.
Hi,
consider the set of the following parametrized matrices
$$
\begin{bmatrix}
1+a & b \\
c & \frac {1 + bc} {1 + a} \\
\end{bmatrix}
$$
They are member of the group ##SL(2,\mathbb R)## (indeed their determinant is 1). The group itself is homemorphic to a quadric in ##\mathbb R^4##.
I believe...
Definitions:
1. A map ##p : X → Y## of smooth manifolds is called a trivial fibration with fiber ##Z## which is also a smooth manifold, if there is a diffeomorphism ##θ : X → Y ×Z## such that ##p## is the composition of ##θ## with the natural projection ##pr_1:Y × Z → Y##.
2. A map ##p: X →Y##...
Given the definition of a smooth map as follows:
A continuous map ##f : X → Y## is smooth if for any pair of charts ##\phi : U →R^m, \psi:V →R^n## with ##U ⊂ X, V ⊂Y##, the map ##\phi(U ∩f^{-1}(V)) → R^n## given by the composition
$$\psi ◦ f ◦ \phi^{-1}$$ is smooth.
Claim: A map ##f : X → Y##...
Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold?
Thank you!
Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$.
I need to prove that the map is differentiable.
But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable?
MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math
Summary:: Books about Kähler manifolds, Ricci flatness and similar things
Hi,
I have an MSc degree in electronics, and I worked with IT. I was always interested in theoretical physics and did extra studies of this in my youth. Now as a pensioner I do private studies for fun. I have studied...
Hello!
According to the attached proposition on ##C^\infty## manifold space of derivations ##D_m M## is isomorphic to Tangent space ##T_m M##.
Cited here another proposition (1.4.5) states the following
1. For constant function ##D_m(f)=0##
2. If ##f\vert_U=g\vert_U## for some neighborhood...
Hi,
I don't know if it is the right place to ask for the following: I was thinking about the difference between the notion of spacetime as 4D Lorentzian manifold and the thermodynamic state space.
To me the spacetime as manifold makes sense from an 'intrinsic' point of view (let me say all the...
I'm having trouble with Rovelli's new book, partly because the info in it is pretty condensed, but also because his subjects are often very different from those in other books on GR like the one by Schutz. For one thing, he never uses the term "manifold", but talks about frame fields, which seem...
According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is:
##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0##
Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...
Hi. I have a question.
Two manifolds can be equivalent but have different co-ordinates and a correspondingly different metric defined on them.
Is there an easy way to identify when two such manifolds are equivalent but just have different co-ordinates?
What do I mean by...
Background:
One can construct a Hilbert space "over" ##\mathbb{R}^{3}## by considering the set of square integrable functions ##\int_{\mathbb{R}^{3}}\left|\psi(\mathbf{r})\right|^{2}<\infty##. That's what is done in QM, and there, even if they are not normalizable, to every...
First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)
I was trying to look for something that works a lot of examples of integrals over surfaces, volumes etc. in general relativity. Tong's notes and some others are good on the abstract/theoretical side but it'd really be better at this stage to get some practice with concrete examples in order to...
I think it could be interesting.
Consider a mechanical system
A circle of mass M can rotate about the vertical axis. The angle of rotation is coordinated by the angle ##\psi##. A bead of mass m>0 can slide along this circle. The position of the bead relative the circle is given by the angle...
Hello. Why do we have different ways of determining curvature on manifolds like the sectional curvature, the scalar curvature, the Riemann curvature tensor , the Ricci curvature? What are their different uses on manifolds? Do they allow each of them different applications on manifolds? Thank you.
Hello. Could we define a topological space that locally resembles a riemannian manifold or another manifold like a complex manifold, or a Hermitian manifold near each point? Could it have interesting properties and theorems? Thank you.
Hi, I just finished up with Riemann Geometry not to long ago, and did something with complex geometry on kahler manifolds. In your opinion what would be a next logical step for someone to study? I am very interested in manifold theory and differential geometry in general. I'm somewhat familiar...
Hi, I have just finished studying Riemannian Geometry and was moving on to trying to figure out what a Kahler manifold is. Using wikipedia's definition(probably a bad idea to start with) it says "Equivalently, there is a complex structure J on the tangent space of X at each point (that is, a...
[Moderator's note: Spin-off from another thread.]
You need the structure of a topological vector field K with 0 as a limit point of K-{0}. The TVF structure allows the addition and quotient expression to make sense; you need 0 as a limit point to define the limit as h-->0 and the topology to...
I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...
Every two semi-Riemannian manifolds of the same dimension, index and constant curvature are locally isometric. If they are also diffeomorphic, are they also isometric?
Hi
I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This...
Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?
Please forgive any confusion, I am not well acquainted with topological analysis and differential geometry, and I'm a novice with regards to this topic.
According to this theorem (I don't know the name for it), we cannot embed an n-dimensional space in an m-dimensional space, where n>m, without...
Hello,
In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as
$$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}')...
Consider two pseudo-Riemmannian manifolds, ##M## and ##N##. Suppose that in coordinates ##y^\mu## on ##M## and ##x^\mu## on ##N##, the Riemann curvatures ##R^M## and ##R^N## of ##M## and ##N## are related by a coordinate transformation ##y = y(x)##:
\begin{equation*}
R^N_{\rho\mu\sigma\nu} =...
Hi there all,
I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...
In the exercises on differential forms I often find expressions such as $$
\omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz
$$ but this is only correct if we're in "flat" space, right?
In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...
Let ##M## be an ##n##-dimensional (smooth) manifold and ##(U,\phi)## a chart for it. Then ##\phi## is a function from an open of ##M## to an open of ##\mathbb{R}^n##. The book I'm reading claims that coordinates, say, ##x^1,\ldots,x^n## are not really functions from ##U## to ##\mathbb{R}##, but...
I'm reading "The Geometry of Physics" by Frankel. Exercise 1.3(1) asks what would be wrong in defining ##||X||## in an ##M^n## by
$$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to...
My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface?
Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...
Hello,
let $$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!
I am in the process of writing a lecture out for my Graduate modeling class I teach. I normally don't write lectures out in LaTex or use PDF's because I write on the dry erase board, but if anyone is interested I wouldn't mind spending the time to type out some notes on the topic.
The topic...
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...