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 3-sphere with Ricci flow

I have a a very basic question and a followup question. 1. Consider you have a 3-sphere, Ricci flow says it contracts to a point in finite time. So the manifold contracts to its center, correct? 2. Say you have two 3-spheres that stay tangent to eachother, and you connect a line between the...
2. I Grassmannian as smooth manifold

Hello! There is a proof that Grassmannian is indeed a smooth manifold provided in Nicolaescu textbook on differential geometry. Screenshots are below There are some troubles with signs in the formulas please ignore them they are not relevant. My questions are the following: 1. After (1.2.5)...
3. I Manifold with boundary

what would the universe look like if its a manifold with boundary? what would it look like at the boundary? and what happens if u try to touch the boundary? is it just a black wall that's unbreakable?
4. I Approximating smooth curved manifolds with "local bits" of curvature?

Consider the electric and magnetic fields around a dipole antenna, Suppose these fields represent some type of curvature in space and time. Suppose where the fields are strong we have greater curvature. Also suppose these fields are really some very large but finite sum of "moving local...
5. I Identification tangent bundle over affine space with product bundle

Hi, as in this thread Newton Galilean spacetime as fiber bundle I'd like to clarify some point about tangent bundle for an Affine space. As said there, I believe the tangent space ##T_pE## at every point ##p## on the affine space manifold ##E## is canonically/naturally identified with the...
6. I Fiber bundle homeomorphism with the fiber

Hi, in the definition of fiber bundle there is a continuous onto map ##\pi## from the total space ##E## into the base space ##B##. Then there are local trivialization maps ##\varphi: \pi^{-1}(U) \rightarrow U \times F## where the open set ##U## in the base space is the trivializing neighborhood...
7. I ##SU(2, \mathbb C)## parametrization using Euler angles

Hi, I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements \begin{pmatrix} e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\ ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}...
8. I Short question about principal bundle

Hello there! Book provides the following definition Let ##(P,G,\Psi)## be a free Lie group action, let ##M## be a manifold and let ##\pi : P \rightarrow M## be a smooth mapping. The tuple ##(P,G,M,\Psi,\pi)## is called a principal bundle, if for every ##m\in M## there exists a local...
9. B Why is the set {(x,y)∈Ω×R|y=f(x)} a manifold?

I am thinking why the following holds: Let f be a smooth function with f: Ω⊂R^m→R. Why is the set {(x,y)∈Ω×R|y=f(x)} a manifold? Would be helpful if you are providing me some guidance or tips:)
10. 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!
11. 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
12. I Clarification about submanifold definition in ##\mathbb R^2##

Hi, a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of...
13. I Is a Manifold with a Boundary Considered a True Manifold?

<Moderator note: thread split from https://www.physicsforums.com/threads/speed-of-light.1012508/#post-6601734 > Is a manifold with a boundary still a manifold?
14. I Darboux theorem for symplectic manifold

Hi, I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem. We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...
15. I Find the center manifold of a 2D system with double zero eigenvalues

I have to find the center manifold of the following system \begin{align} \dot{x}_1&=x_2 \\ \dot{x}_2&=-\frac{1}{2}x_1^2 \end{align} which has a critical point at ##x_0=\begin{bmatrix}0 & 0\end{bmatrix}##. Its linearization at that point is \begin{align} D\mathbf {f}(\mathbf {x_0}) =...
16. 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?)
17. A String Theory in N dimensions?

String Theory and related theories like M Theory have strong constraints in the number of dimensions where they can be formulated (for example, in the case of M theory, it is only allowed in 11D or in the case of bosonic string theory is only allowed in 26D. Since string theory and related...
18. I GTR & STR: Could 9 Dimensions Unify Theory?

Let's play pretend a progressive alien civilisation contacts us and an irrelevant conversation begins. Later on, an alien-scientist says: "by the way, the physical reality contains 9 dimensions. I heard a famous human theorist announced, that it should be 4. You have to touch up." Could the GTR...
19. A Name for a subset of real space being nowhere a manifold with boundary

I was wondering if anyone knew of a name for such a set, namely a subset S \subseteq \mathbb{R}^n which at every point x \in S there exists no open subset U of \mathbb{R}^n containing x such that S \cap U is homeomorphic to either \mathbb{R}^m or the half-space \mathbb{H}^m = \{(y_1,...,y_m)...
20. I Is H a Lie Group with Subspace Topology from T^2?

"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0...
21. I What is an (almost) complex manifold in simple words

I try to understand (almost) complex manifolds and related stuff. Am I right that the condition for almost complexity simply is that the metric locally can be written in terms of the complex coordinates ##z##, i.e. ##g = g(z_1, ... z_m)## (complex conjugate coordinates must not appear)? These...
22. B Increasing the dimensions of a manifold

Suppose I have a R^3 manifold that goes into R^3 charts, if that is possible. The manifold has curvature and is Riemannian and has a metric. I want to eliminate all curvature in R^3 charts, so I want to add another dimension to the manifold, I would extract all the curvature information from the...
23. B Topology on flat space when a manifold is locally homeomorphic to it

[I urge the viewer to read the full post before trying to reply] I'm watching Schuller's lectures on gravitation on youtube. It's mentioned that spacetime is modeled as a topological manifold (with a bunch of additional structure that's not relevant to this question). A topological manifold is...
24. A Question about the derivation of the tangent vector on a manifold

I am trying to understand the following derivation in my lecture notes. Given an n-dimensional manifold ##M## and a parametrized curve ##\gamma : (-\epsilon, \epsilon) \rightarrow M : t \mapsto \gamma(t)##, with ##\gamma(0) = \mathbf{P} \in M##. Also define an arbitrary (dummy) scalar field...
25. I A set of numbers as a smooth curved changing manifold.

Edit, the vector that rotates below might not rotate at all. Please forgive any mistaken statements or sloppiness on my part below. I think that by some measure a helicoid can be considered a smooth curved 2 dimensional surface except for a line of points? Consider not the helicoid above...
26. I Why Isn't the Intersection of Two Lines a 1D Manifold?

This is a very simple topology question. Consider two infinite lines crossing at one point. Now, I know that this is not a 1D manifold, and I know the usual argument (in the neighbourhood of the intersection, we don't have a a line, or that if we remove the intersection point, we end up with...
27. B Size Manifold Globally: Euclidean 3D Space

I was wondering if it was possible to determine the size of a manifold globally. Suppose I had a manifold that sits in 3 dimensions. I could construct a Euclidean space around in the same space and be able to say things of the dimensions right?
28. 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...
29. I Defining a Point on a Manifold: Intrinsic vs Embedded Space

Say you have some n dimensional manifold embedded in a higher space. what is the best way to describe or define a point on a manifold with or without coordinates. How could I do this either intrinsically or using the embedded space. Would you use the tangent space somehow using basis vectors?
30. I Trying to construct a particular manifold locally using a metric

I am trying to construct a particular manifold locally using a metric, Can I simply take the inner product of my basis vectors to first achieve some metric.
31. I About the definition of a Manifold

Hi, I'm a bit confused about the locally euclidean request involved in the definition of manifold (e.g. manifold ): every point in ##X## has an open neighbourhood homeomorphic to the Euclidean space ##E^n##. As far as I know the definition of homeomorphism requires to specify a topology for...
32. I What's the difference between graph, locus & manifold?

They all seem to mean the same thing. I personally have been using locus.
33. 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...
34. I Is There a Generalized Fourier Transform for All 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?

37. I Are Coordinates on a Manifold Really Functions from R^n to R?

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...

48. A Constructing a sequence in a manifold

Given S is a submanifold of M such that every smooth function on S can be extended to a smooth function to a neighborhood W of S in M. I want to show that S is embedded submanifold. My attempt: Suppose S is not embedded. Then there is a point p that is not contained in any slice chart. Since a...
49. I Is the Boundary Chart for a Closed Unit Ball Injective?

I want to show that a closed unit ball is manifold with boundary and I attempted as uploaded. But I am not happy with the way I showed the boundary chart is injective. Am I right?
50. I Calabi-Yau manifold + ideal gas + point disturbance?

Because it is a closed space, can it make sense to fill a Calabi_Yau manifold with an ideal gas and consider waves from a point disturbance? Would the Ricci-flat condition of Calabi-Yau manifolds have anything to say about possible sound waves? Thanks!