# What is Manifolds: Definition and 282 Discussions

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.

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1. ### I ##SL(2,\mathbb R)## Lie group as manifold

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...
2. ### A Regarding fibrations between smooth manifolds

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##...
3. ### A A claim about smooth maps between smooth manifolds

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##...
4. ### I Are the coordinate axes a 1d- or 2d-differentiable manifold?

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!
5. ### I Is the projective space a smooth manifold?

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
6. ### Books about Kähler manifolds, Ricci flatness and other things

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...
7. ### I Understanding Derivations and Tangent Spaces on Manifolds

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...
8. ### I Comparing Spacetime and Thermodynamic State Space Manifolds

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...
9. ### A Understanding Frame Fields in GR: A Beginner's Guide

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...
10. ### I Parallel Transport of a Tensor: Understand Equation

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...
11. ### I Identifying Manifolds with Different Co-ordinates: Easy Solutions

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...
12. ### A Hilbert spaces and kets "over" manifolds

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...
13. ### A Differential forms on R^n vs. on manifold

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?)
14. E

### Calculus Practical reference for integration on manifolds

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...
15. ### A Something about configuration manifolds in classical mechanics

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...
16. T

### A On the different ways of determining curvature on manifolds

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.
17. ### Carroll chapter 2 questions 9 and 10 on Manifolds

Summary:: hodge Duality... Does anybody has idea for solving these 2 problems?
18. T

### B Examples of manifolds not being groups

Hello there. Do you know any examples of manifolds not being groups?Can you talk about some of them developing them as much as you want?Thank you.
19. T

### I Exploring Topological Spaces Resembling Manifolds: Properties and Theorems

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.
20. ### I Lorentz Invariance Violation for Manifolds

I was looking at this video , and I was wondering if a (Riemannian)manifold violates the "lorentz invariance" would it become a discrete manifold?
21. ### Geometry I would like suggestions regarding reading about geometry and manifolds

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...
22. ### A Understanding Kahler Manifolds

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...
23. ### I Differentiable manifolds over fields other than R, C

[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...
24. ### I How do charts on differentiable manifolds have derivatives without a metric?

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...
25. ### A Diffeomorphic manifolds of equal constant curvature

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?
26. ### A Logical foundations of smooth manifolds

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...
27. ### I Fourier transform on manifolds

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?
28. ### I Embedding homeomorphic manifolds

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...

38. ### Spacetime and Geometry: Vanishing commutators#2

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...
39. ### Spacetime and Geometry: Vanishing commutators

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...
40. ### I Munkres-Analysis on Manifolds: Extended Integrals

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...
41. ### I Munkres-Analysis on Manifolds: Theorem 20.1

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)...
42. ### A Penrose paragraph on Bundle Cross-section?

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...
43. ### A Reshetikhin-Turaev Invariant of Manifolds

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...
44. ### A Can Smooth Functions be Extended on Manifolds?

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...
45. ### B String theory, Calabi–Yau manifolds, complex dimensions

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!
46. ### MHB Gathering interest in Dynamical Systems (Bifurcation, Stable Manifolds, Hamiltonian Systems) f

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...
47. ### A On the dependence of the curvature tensor on the metric

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...
48. ### A Can you give an example of a non-Levi Civita connection?

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...
49. ### A Questions about Covering maps, manifolds, compactness

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...
50. ### A Connected sum of manifolds and free group isomorphisms

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...