Manifolds Definition and 290 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. cianfa72

    I Notion of congruent curve along a vector field

    Consider the following: suppose there is a smooth vector field ##X## defined on a manifold ##M##. Take a smooth curve ##\alpha(\tau)## between two different integral curves of ##X## where ##\tau## is a parameter along it. Let ##A## and ##B## the ##\alpha(\tau)## 's intersection points with the...
  2. cianfa72

    I Show that a "cross" is not a topological manifold

    Hi, I've a doubt about the following example in "Introduction to Manifold" by L. Tu. My understanding is that if one assumes the subspace topology from ##\mathbb R^2## for the "cross", then one can show that the topological space one gets is Hausdorff, second countable but non locally...
  3. cianfa72

    I SL(n,R) Lie group as submanifold of GL(n,R)

    Hi, consider the group ##SL(n,\mathbb R)##. It is a subgroup of ##GL(n,\mathbb R)##. To show it is a Lie group we must assign a differential structure turning it into a differential manifold, proving further that multiplication and taking the inverse are actually smooth maps. With the...
  4. cianfa72

    I 2-sphere manifold intrinsic definition

    Hi, in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space ##\mathbb R^3##. On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts. I believe the same view might also be...
  5. Onyx

    I Correct Description of Black Hole Interior

    Does the interior coordinate patch of the Schwarzschild analytic extension really describe the interior of a black hole? After all, that portion would have mass. Also, is there a way to describe just a black hole’s with regular spherical coordinates?
  6. cianfa72

    I Must a Smooth Section Over a Mobius Strip Take Value Zero at Some Point?

    As discussed in a recent thread, I'd ask whether any smooth section over a Mobius strip must necessarily take value zero on some point over the base space ##\mathbb S^1##. Edit: my doubt is that any closed curve going in circle two times around the strip is not actually a section at all. Thanks.
  7. cianfa72

    I Spacetime coordinate smoothness requirement

    Hi, I was keep reading the interesting book Exploring Black Holes - second edition from Taylor, Wheeler, Bertschinger. I'd like to better understand some points they made. In Box 3 section 3-6 an example of coordinate singularity at point O in Euclidean plane in polar coordinates centered there...
  8. PhysicsRock

    I Definition of manifolds with boundary

    In differential geometry, we typically define the boundary ##\partial M## of a manifold ##M## as all ##p \in M## for which there exists a chart ##(U,\varphi), p \in U## such that ##\varphi(p) \in \partial\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n = 0 \}##. Consequently, we also demand that...
  9. cianfa72

    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...
  10. A

    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##...
  11. A

    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##...
  12. 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!
  13. 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
  14. StenEdeback

    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...
  15. K

    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...
  16. cianfa72

    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...
  17. joneall

    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...
  18. A

    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...
  19. W

    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...
  20. A

    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...
  21. 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?)
  22. 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...
  23. wrobel

    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...
  24. 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.
  25. Pouramat

    Carroll chapter 2 questions 9 and 10 on Manifolds

    Summary:: hodge Duality... Does anybody has idea for solving these 2 problems?
  26. 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.
  27. 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.
  28. S

    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?
  29. S

    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...
  30. S

    A What Is a Kahler Manifold?

    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...
  31. W

    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...
  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. C0nstantine

    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?
  34. 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...
  35. 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?
  36. Q

    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...
  37. 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}')...
  38. T

    A Distinguishing Riemannian Manifolds by Curvature Relationships

    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} =...
  39. A

    Calculus Multivariable calculus without forms or manifolds

    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...
  40. K

    I Understanding Differential Forms and Basis Vectors in Curved Space

    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...
  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. K

    I Bijectivity of Manifolds: Can m≠n?

    Is it possible for a manifold to be homeomorphic to ##R^m## in some regions and homeomorphic to ##R^n## in other regions, with ##m \neq n##?
  43. 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...
  44. P

    A Fundamental definition of extrinsic curvature

    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...
  45. M

    A Differentiability of a function between manifolds

    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...
  46. George Keeling

    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...
  47. George Keeling

    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...
  48. Bill2500

    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...
  49. Bill2500

    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)...
  50. Abhishek11235

    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...
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