What is Manifold: Definition and 327 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. J

    A Is tangent bundle TM the product manifold of M and T_pM?

    Hello. I was trying to prove that the tangent bundle TM is a smooth manifold with a differentiable structure and I wanted to do it in a different way than the one used by my professor. I used that TM=M x TpM. So, the question is: Can the tangent bundle TM be considered as the product manifold...
  2. fresh_42

    Insights A Journey to The Manifold SU(2) - Part II - Comments

    Greg Bernhardt submitted a new PF Insights post A Journey to The Manifold SU(2) - Part II Continue reading the Original PF Insights Post.
  3. P

    I Why must the spacetime we inhabit be a geodesically complete manifold?

    Can someone tell me how we know that our physical universe is geodesically complete? In response to a question I had about why we assign any meaning to the other side of a black hole’s event horizon (or its interior), I got an answer prompting me to look into the concept of geodesic...
  4. fresh_42

    Insights What Defines a Local Lie Group in A Journey to The Manifold - Part I?

    fresh_42 submitted a new PF Insights post A Journey to The Manifold - Part I Continue reading the Original PF Insights Post.
  5. A

    I What is an atlas for a torus manifold?

    I don't quite understand some work I'm doing creating the normal Riemann surface for the function ##f(z)=\frac{A}{\sqrt{(1-z^2)(k^2-z^2)}}##. I can use Schwarz-Christoffel transforms to map the function to a rectangular polygon in the zeta-plane then map this rectangle onto a torus. But I...
  6. PsychonautQQ

    I Can a Topological Space be Both a 1-Manifold and an n-Manifold?

    This is not homework, self study, so I'm going to post here, where people know things :P. a) Show that a topological space can't be both a 1-manifold and an n-manifold for any n>1. If a topological space were both a 1-manifold and an n-manifold, then every point would have a neighborhood...
  7. S

    I Understanding Integration on an Orientable Manifold

    Hello! I am reading how to integrate on an orientable manifold. So we have ##f:M \to R## and an m-form (m is the dimension of M): ##\omega = h(p)dx^1 \wedge ... \wedge dx^m##, where ##h(p)## is another function on the manifold which is always positive as the manifold is orientable. The way...
  8. S

    I Lie bracket on a manifold

    Hello! So I have 2 vector fields on a manifold ##X=X^\mu\frac{\partial}{\partial x^\mu}## and ##Y=Y^\mu\frac{\partial}{\partial x^\mu}## and this statement: "Neither XY nor YX is a vector field since they are second-order derivatives, however ##[X, Y]## is a vector field". Intuitively makes...
  9. S

    I What are the components of a vector field on a manifold?

    Hello! I am not sure I understand the idea of vector field on a manifold. The book I read is Geometry, Topology and Physics by Mikio Nakahara. The way this is defined there is: "If a vector is assigned smoothly to each point on M, it is called a vector field over M". Thinking about the 2D...
  10. davidge

    I Is it possible to prove that the circle is a manifold using open spheres?

    The answer to the question of the thread title is yes, according to what I found on web. Now a manifold is by definition a topological space that (aside from other conditions) is locally Euclidean. What does such condition means? Is it the same as saying that each of its points must have a...
  11. R

    [Symplectic geometry] Show that a submanifold is Lagrangian

    Homework Statement Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...
  12. M

    B Exploring Manifold Duplication in Many Worlds: The Mathematical Expression

    What's the mathematical expression for manifold duplication in Many Worlds? In other words, how do the manifolds duplicate themselves endlessly?
  13. davidge

    I Vectors and isometries on a manifold

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  14. Heisenberg1993

    A Two cones connected at their vertices do not form a manifold

    Why is i that two cones connected at their vertices is not a manifold? I know that it has to do with the intersection point, but I don't know why. At that point, the manifold should look like R or R2?
  15. A

    A Physical meaning of open set on manifold

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  16. FallenApple

    I How is a manifold locally Euclidean?

    So if I pick any 2 points on a 2d manifold, say x1 and x2, then the distance between these two points is a secant line that passes through 3 space that isn't part of the manifold. So no matter what, there doesn't exist an point epsilon, e , where ||e ||>||0 || and ||x2-x1||<|| e || No matter...
  17. D

    I How do I calculate the surface area of a rotated curve?

    How do I find the surface area of a f(x) rotated around the y axis?
  18. S

    Parameterized p-subset of a manifold M

    Homework Statement A parameterized ##p##-subset ##(U,F)## of a manifold ##M^{n}## is ''irregular'' at ##u_0## if rank ##F<p## at ##u_0##. Show that if ##\alpha^{p}## is a form at such a ##u_0## then ##F^{*}\alpha^{p}=0##. Homework Equations The Attempt at a Solution A parameterized...
  19. T

    I How does parallel transportation relates to Rieman Manifold?

    Source: Basically the video talk about how moving from A to A'(which is basically A) in an anticlockwise manner will give a vector that is different from when the vector is originally in A in curved space. $$[(v_C-v_D)-(v_B-v_A)]$$ will equal zero in flat space...
  20. orion

    I Do derivative operators act on the manifold or in R^n?

    I am really struggling with one concept in my study of differential geometry where there seems to be a conflict among different textbooks. To set up the question, let M be a manifold and let (U, φ) be a chart. Now suppose we have a curve γ:(-ε,+ε) → M such that γ(t)=0 at a ∈ M. Suppose further...
  21. orion

    I Question about chart parameterization

    Suppose we have an n-dimensional manifold Mn and take a coordinate neighborhood U with associated coordinate map φ: U → V where V is an open subset of ℝn. So far I'm clear on this. However, where I become confused is when some books say that φ-1 is called a parameterization of U and basically...
  22. E

    I Bounding the volume distortion of a manifold

    Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection. Consider the manifold $M=f(U)$. Its volume distortion is defined as $G=det(DftDf).$ If $n=1$, one can deduce that $G=1+|\nabla f|^2$. What happens for $n>1$? Can one bound from below this $G$? If...
  23. S

    A Concept of duality for projective spaces and manifolds

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  24. G

    A Mapping Tangent Space to Manifold - Questions & Answers

    Hi all, this might be a silly question, but I was curious. In Carroll's book, the author says that, in a manifold M , for any vector k in the tangent space T_p at a point p\in M , we can find a path x^{\mu}(\lambda) that passes through p which corresponds to the geodesic for that...
  25. F

    A Continuous map from one manifold to another

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  26. C

    Can a non-local manifold coexist with a spacetime manifold?

    Is it possible to create a non-local manifold that co exist with the spacetime manifold? The non-local manifold being where quantum correlations took place. How do you make the two manifolds co-exist?
  27. aboutammam

    A Transversality unlocks the secrets of the manifold?

    Hi everybody I seen in a book a quote of H. E. Winkelnkemper "Transversality unlocks the secrets of the manifold.". Can someone explain to me this citation and why transversality has all this importance in geometry? thank you in advance.
  28. J

    Curvature of a group manifold

    So I'm working with some group manifolds. The part that's getting to me is the Ricci scalar I'm using to describe the curvature. I have identified the groups that I'm using but that's not really relevant at the moment. We're using a left-invariant metric ##\mathcal{M}_{ab}##. Now I've got the...
  29. N

    Spivak & Dimension of Manifold

    1. Homework Statement I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the...
  30. J

    N-sphere as manifold without embedding

    I have been working through John Lee's "Introduction to Smooth Manifolds" recently. I am having trouble visualizing the unit n-sphere as a manifold in its own right, i.e. without an ambient space. It seems impossible to visualize it without embedding it in Euclidean space. I mean, the unit...
  31. C

    Why is a branched line in R2 not a topological manifold?

    Is there a topologist out there that wants to explain why exactly a branched line in R2 is not not a topological manifold? I know it's because there doesn't exist a chart at the point of branching, but I don't understand why not. I'm just starting to self study this, so go easy on me :).
  32. D

    Tangent spaces at different points on a manifold

    Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...
  33. D

    Optimizing Flow in a Networked Manifold System

    Hi, A chamber (manifold type cylinder) has 1 inlet and 8 outlets of 3 inch diameter each. 8 suction blowers are connected to the chamber's outlet. Each blower's suction flow rate is 1000 cfm. what will be the flow rate through the inlet of the chamber (diameter = 3 inches). Will the inlet...
  34. W

    Writing a general curve on a manifold given a metric

    I have what I think is a basic question. Say I have a manifold and a metric. How do I write down the most general curve for some arbitrary parameter? For example in \mathbb{R}^2 with the Euclidean metric, I think I should write \gamma(\lambda) = x(\lambda)\hat{x} + y(\lambda)\hat{y} But what...
  35. S

    ##x+y## on a Riemannian manifold

    Can one define a vector space structure on a Riemannian manifold ##(M,g)##?! By this I mean, does it make a sense to write ##x+y## where ##x,y## are arbitrary points on ##M##?
  36. J

    Center manifold and submanifold

    Hi all, I am not familiar with the dynamic system theory. When I was trying to understand the weakly nonlinear stability analysis, I realize the following question. It is known that the center manifold reduction can be used to study the first linear bifurcation. This lead to the...
  37. D

    Geodesic Transport of Small 2D Surface on 3D Manifold

    Hello, I've just read and I think I have understood the following result : If we were to geodesically transport all points of a small 2D surface, so small that it would be flat for all purposes, in a direction vertically above it, and if this surface belongs in an arbitrary 3D manifold, then in...
  38. S

    Why do diffeomorphic manifolds have physically identical properties?

    Let f:p\mapsto f(p) be a diffeomorphism on a m dimensional manifold (M,g). In general this map doesn't preserve the length of a vector unless f is the isometry. g_p(V,V)\ne g_{f(p)}(f_\ast V,f_\ast V). Here, f_\ast:T_pM\to T_{f(p)}M is the induced map. In spite of this fact why...
  39. D

    Local parameterizations and coordinate charts

    I have recently had a lengthy discussion on this forum about coordinate charts which has started to clear up some issues in my understanding of manifolds. I have since been reading a few sets of notes (in particular referring to John Lee's "Introduction to Smooth Manifolds") and several of them...
  40. jford1906

    Vector fields transverse to the boundary of a manifold

    I'm trying to work up some examples to help me understand this concept. Would the periodic flow on a solid torus be transverse to it's boundary?
  41. Ravi Mohan

    Single chart which covers entire [itex]S^1\times R[/itex] manifold

    I am studying Carroll's notes on GR. He defines charts as maps from an open subset in manifold M to open subset in R^n. He then writes "We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system. (Although some can, even ones with...
  42. D

    Coordinate charts and change of basis

    So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...
  43. B

    Exploring the Differences: Manifold vs. Submanifold

    Hello, (Continuing learner: new to all of this: patience requested, please.) What is the difference between a manifold and a submanifold? I have read and now been working with each. I understand the technical definitions (I think I do... but I do not have an intrinsic FEELING for these...
  44. D

    Differential map between tangent spaces

    I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...
  45. H

    Intake manifold pressure and spark timing

    Why does higher intake manifold pressure results in increase in the degree of spark advancement? Similarly, why is the torque higher for higher intake manifold pressure?
  46. C

    Curves and tangent vectors in a manifold setting

    Consider the following definition: (##M## denotes a manifold structure, ##U## are subsets of the manifold and ##\phi## the transition functions) Def: A smooth curve in ##M## is a map ##\gamma: I \rightarrow M,## where ##I \subset \mathbb{R}## is an open interval, such that for any chart...
  47. D

    Integrating Forms on Manifolds: Understanding the Concept and Techniques

    In all the notes that I've found on differential geometry, when they introduce integration on manifolds it is always done with top forms with little or no explanation as to why (or any intuition). From what I've manage to gleam from it, one has to use top forms to unambiguously define...
  48. F

    Understanding Manifolds: Questions & Answers

    I'm just trying to understand something about manifolds. What is meant when a manifold doesn't have boundary? I thought the boundary was where the manifold "ends" so to speak. Like a boundary point, something where you take a small nhbd (neighborhood) and you get something inside the set and...
  49. D

    Why are vectors defined in terms of curves on manifolds

    What is the motivation for defining vectors in terms of equivalence classes of curves? Is it just that the definition is coordinate independent and that the differential operators arising from such a definition satisfy the axioms of a vector space and thus are suitable candidates for forming...
  50. D

    Attempting to understand diffeomorphisms

    I am relatively new to the concept of differential geometry and my approach is from a physics background (hoping to understand general relativity at a deeper level). I have read up on the notion of diffeomorphisms and I'm a little unsure on some of the concepts. Suppose that one has a...
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