Manifold Definition and 335 Threads

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. H

    A Vector Spaces Associated with Quark Modes in k-Space

    My idea is as follows. Each mode of the quark field is characterized by a wave vector k. Each wave vector corresponds to a point in k-space. This set of points representing different modes forms a manifold. Each point in k-space can be assigned a three-dimensional vector space that represents...
  2. Onyx

    I Metric Tensor on ##S^1## x ##S^2##

    How do I find the metric tensor on ##S^1## x ##S^2##?
  3. L

    I Definition of a Manifold

    Why do we need second countable and Hausdorff conditions for manifold definition?
  4. kappaka

    A Understanding: double of conformally flat manifold is conformally flat

    I'm reading a paper and there is a proof that the double of a compact locally conformally flat Riemannian manifold with totally geodesic boundary again carries a locally conformally flat structure. The proof is as follows: Let \( (M^n, g) \) be a locally conformally flat compact manifold with...
  5. casparov

    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...
  6. K

    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)...
  7. DDTG Global

    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?
  8. Spinnor

    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...
  9. cianfa72

    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...
  10. cianfa72

    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...
  11. cianfa72

    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}}...
  12. K

    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...
  13. SaschaSIGI

    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:)
  14. D

    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!
  15. D

    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
  16. cianfa72

    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...
  17. Dale

    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?
  18. cianfa72

    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...
  19. N

    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}) =...
  20. K

    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?)
  21. S

    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...
  22. E

    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...
  23. D

    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)...
  24. Y

    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...
  25. G

    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...
  26. S

    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...
  27. S

    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...
  28. Decimal

    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...
  29. Spinnor

    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...
  30. N

    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...
  31. S

    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?
  32. J

    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...
  33. D

    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?
  34. D

    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.
  35. cianfa72

    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...
  36. S

    I What's the difference between graph, locus & manifold?

    They all seem to mean the same thing. I personally have been using locus.
  37. Avatrin

    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...
  38. redtree

    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?
  39. A

    I Tangent vector basis and basis of coordinate chart

    I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...
  40. ZuperPosition

    Abstract definition of electromagnetic fields on manifolds

    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}')...
  41. K

    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...
  42. E

    A Lie derivative of vector field defined through integral curv

    Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted ##\phi _ { t } ( p ) .## Now consider $$t \mapsto a _ { t } \left( \phi _ { t } (...
  43. K

    I Gradient vector without a metric

    Is it possible to introduce the concept of a gradient vector on a manifold without a metric?
  44. cianfa72

    I Injective immersion that is not a smooth embedding

    Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8 ##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)## As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...
  45. N

    A Study Chern-Simons Invariant: Understanding 3-Manifold Measurement

    I've been studying the Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds but have almost zero background in physics. The WRT of a 3-manifold is closely related to the Chern-Simons (CS) invariant via the volume conjecture. My question is, what does the CS invariant of a 3-manifold...
  46. K

    A Intrinsic definition on a manifold

    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...
  47. cianfa72

    B Differentiable function - definition on a manifold

    Hi, a basic question related to differential manifold definition. Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined...
  48. R

    A Pullback of F on Manifolds: What Matrix Do We Take Determinant Of?

    Hey, we had in the lecures the following: Let M and N be smooth manifolds, and dim(M)=dim(N)=n, while $$x^i$$ and $$ y^i$$ are coordinate functions around $$p\in M$$ respective $$F(p) \in N$$, then we get for the pullback of F Which entries has the matrix we take the determinant of? I thaught...
  49. W

    I Parametrization manifold of SL(2,R)

    I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a...
  50. shahbaznihal

    A On metric and connection independence

    Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast. I will be very thankful if...
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