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. K

    What is the meaning of constant on each others fibres in differential geometry?

    This should hopefully be a quick and easy answer. I'm running through Lee's Introduction to Smooth Manifolds to brush up my differential geometry. I love this book, but I've come to something I'm not sure about. He states a result for which the proof is an exercise: I'm not quite clear on...
  2. B

    Example of Codimension-1 Manifolds that are not Isotopic.?

    Hi, Everyone: A question on knots, please; comments,references appreciated. The main points of confusion are noted with a ***: 1)I am trying to understand how to describe the knot group Pi_1(S^3-K) as a handlebody ( this is not the Wirtinger presentation; this is from some...
  3. mnb96

    Quaternions and associated manifolds

    Hello, it is known that pure-quaternions (scalar part equal to zero) identify the \mathcal{S}^2 sphere. Similarly unit-quaternions identify points on the \mathcal{S}^3 sphere. Now let's consider quaternions as elements of the Clifford algebra \mathcal{C}\ell_{0,2} and let's consider a...
  4. K

    Descent directions on manifolds

    Hey, I'm trying to do some optimization on a manifold. In particular, the manifold is \mathfrak U(N) , the NxN unitary matrices. Now currently, I'm looking at "descent directions" on the manifold. That is, let f: \mathfrak U(N) \to \mathbb R be a function that we want to minimize, p...
  5. D

    Spivak (Calculus on Manifolds) proof of stolkes theorem

    http://planetmath.org/?op=getobj&from=objects&id=4370 that's pretty much the proof of Stolkes Theorem given in Spivak but I'm having a lot of difficulty understanding the details specifically...when the piecewise function is defined for j>1 the integral is 0 and for j=1 the integral is...
  6. O

    Ricci form and Kahler manifolds

    I am confused about the different Ricci-named objects in complex and specifically Kahler geometry: We have the Ricci curvature tensor, which we get by contracting the holomorphic indices of the Riemann tensor. We have the Ricci scalar Ric, which we get by contracting the Ricci tensor. Then there...
  7. B

    Orientability of Complex Manifolds.

    Hi, everyone: I am trying to show that any complex manifold is orientable. I know this has to see with properties of Gl(n;C) (C complexes, of course.) ; specifically, with Gl(n;C) being connected (as a Lie Group.). Now this means that the determinant map must be either...
  8. N

    Easy-to-Follow Proofs for Symplectic Manifolds: A Comprehensive Resource

    What's a really good resources with numerous easy-to-follow proofs to theorems on symplectic manifolds? Arnold is too difficult.
  9. P

    Differentiation on Euclidean Space (Calculus on Manifolds)

    Homework Statement This is from Spivak's Calculus on Manifolds, problem 2-12(a). Prove that if f:Rn \times Rm \rightarrow Rp is bilinear, then lim(h, k) --> 0 \frac{|f(h, k)|}{|(h, k)|} = 0 Homework Equations The definition of bilinear function in this case: If for x, x1, x2...
  10. N

    Differentiating Composition of Smooth Functions

    Homework Statement Let f: M \rightarrow N , g:N \rightarrow K , and h = g \circ f : M \rightarrow K . Show that h_{*} = g_{*} \circ f_{*} . Proof: Let M, N and K be manifolds and f and g be C^\infinity functions. Let p \in M. For any u \in F^{\infinity}(g(f((p))) and any...
  11. M

    Divergence Theorem on Manifolds

    Hi, I'm having some trouble understanding this theorem in Lang's book, (pp. 497) "Fundamentals of Differential Geometry." It goes as follows: \int_{M} \mathcal{L}_X(\Omega)= \int_{\partial M} \langle X, N \rangle \omega where N is the unit outward normal vector to \partial M , X...
  12. arivero

    List of Compact 7 dimensional Einstein manifolds

    The most recent version of the theorem, as stated by Nikonorov in 2004 Let G a connected compact semisimple Lie group, which acts almost effectively on a seven-dimensional simply-connected homogeneous space M^7=G/H. If (G/H,\rho) is a homogeneous Einstein manifold, then it is either a symmetric...
  13. P

    Conformally flat manifolds

    Why all two dimensional manifolds are conformally flat? Why all manifolds with constant sectional curvature are conformally flat? Does anyone know proofs of above statements. Thanks in advance.
  14. Spinnor

    Embedding manifolds that are not very flat.

    I think I read on these forums that a small piece of relatively flat 4D spacetime of General Relativity can be embedded in ten dimensions. What happens if the small piece of spacetime is not very flat, does this change the number of embedding dimensions required? Thanks for any help!
  15. mnb96

    Geodesics in non-smooth manifolds

    Hello, I will expose a simplified version of my problem. Let's consider the following transformation of the x-axis (y=0) excluding the origin (x\neq 0): \begin{cases} \overline{x}=x \\ \overline{y}=1/x \end{cases} Now the x-axis (excluding the origin) has been transformed into an hyperbola...
  16. R

    Unique Distance on Pseudo-Riemannian Manifolds: Riccardo's Question

    What conditions do we have to put on a pseudo-Riemannian manifold in order for a unique and well defined concept of distance between events to be meaningful? I'm thinking about for example max. length of geodesic connecting two events. We have to require one such maximum or minimum length to...
  17. C

    Relationship between Chern and Levi-Civita Connections on Kahler Manifolds

    So I'm trying to understand the statement: On a complex manifold with a hermitian metric the Levi-Civita connection on the real tangent space and the Chern connection on the holomorphic tangent space coincide iff the metric is Kahler. I basically understand the meaning of this statement, but...
  18. R

    PDEs, Manifolds & Frobenius: Intro Course Insights

    I'm looking to delve into PDEs. I'm reading thru Lee's Smooth Manifolds, and he has a chapter on integral manifolds, and how they relate to PDE solutions via Frobenius' theorem. I find the hint of geometrical aspects very appealing. Evans' PDE book (that I was planning on picking up)...
  19. honestrosewater

    Relationship between manifolds and random variables

    I am studying calculus and statistics currently, and a possible relationship between them just occurred to me. I was thinking about two things: (i) is a differentiable function from R to R a manifold, and (ii) in what way is a random event really unpredictable? So I don't know much about either...
  20. S

    Computing tangent spaces of implicitly defined manifolds

    Hi there, Is there an "easy" way to find a tangent space at a specific point to an implicitly defined manifold? I am thinking of a manifold defined by all points x in R^k satisfying f(x) = c for some c in R^m. Sometimes I can find an explicit parametrization and compute the Jacobian matrix...
  21. M

    Reading Analysis on Manifolds by Munkres

    Show that U(x0, ε) is an open set. I'm reading Analysis on Manifolds by Munkres. This question is in the review on Topology section. And I've just recently been introduced to basic-basic topology from Principles of Mathematical Analysis by Rudin. I'm not really certain where to begin...
  22. V

    Strings & Manifolds: Do Strings Vibrate in 10-11D?

    In string theory are the strings themselves the manifolds? or are the strings vibrating in a 10 or 11-d manifold?.
  23. BWV

    Are there operators that change the curvature of manifolds?

    Seen a couple of pieces on gauge theories of finance equating arbitrage opportunities to curvature, for this to work the curvature must therefore not be constant, instead it would be some sort of a function of capital flows
  24. L

    Question about topological manifolds

    Does every (continous and second countable) topological manifold have an Euclidean neighbourhood around each of its points whose closure equals the whole manifold?
  25. E

    Cartesian product of orientable manifolds

    The problem is to prove that if M and N are orientable manifolds, then MxN is an orientable manifold
  26. B

    Spivak Calculus on Manifolds

    Homework Statement Given a Jordan-measurable set in the yz-plane, use Fubini's Thm to derive an expression for the volume of the set in R3 obtained by revolving the set about the z-axisHomework Equations The Attempt at a Solution I solved this problem very easily using change of variable...
  27. T

    Finding a Book on Manifolds: Definitions and More

    Please: I need abook that include this Definitions: 1- Manifold 2- Stable Manifold 3- unstable Manifold thank you.
  28. B

    Handles and non-orientible 3D manifolds

    I've read that with 2D manifolds, you can create any closed 2D manifold by adding "handles" or "crosscaps" (or "crosshandles"). To add a handle, cut out two disks and add the ends of a circle x interval product (cylinder). If you glue one end "the wrong way" you get a "crosshandle", which is how...
  29. B

    Spivak's calculus on manifolds

    I am currently working through spivak's calculus on manifolds (which i love by the way) in one of my class. my question is about his notation for partial derivatives. i completely understand why he uses it and how the classical notation has some ambiguity to it. however, i can't help but...
  30. P

    Manifolds and Lorentz-group

    Homework Statement I've got a problem. I should discribe all minimal invariant manifolds in Minkowski-space, where the proper Lorentz-group \mathcal{L}_{+}^{\uparrow} acts transitivly (i.e. any two points of the manifold can be transformed into each other by a Lorentz-transformation)...
  31. W

    Spivak's Calculus on Manifolds?

    What are some preliminary texts/knowledge before approaching: Spivak's Calculus on Manifolds?
  32. D

    Differentiation on Smooth Manifolds without Metric

    Hi, I'm confused about what differentiation on smooth manifolds means. I know that a vector field v on a manifold M is a function from C^{\infty}(M) to C^{\infty}(M) which is linear over R and satisfies the Leibniz law. This should be thought of, I'm told, as a 'derivation' on smooth...
  33. Jim Kata

    Something I read somewhere about Spin manifolds, I don't remember where?

    What I'm about to say is really sketchy since I don't even remember where I read this, and don't really understand the topic, but I thought it was cool. Basically, it said on a spin manifold, a manifold where can do a lift from, for example an SO(3) twisted bundle to Spin(3), the weights of...
  34. K

    What comes first, spivaks manifolds or rudins pma?

    im buying books to get better at proofs so that i can tackle rudins analysis text. my question is, do i read spivak manifolds before or after rudin? a lot of sources list manifolds as a second year text, suggesting it is required reading. but some people also say it should be read after or...
  35. S

    Visualising calabi yau manifolds

    I would like to make visualisations of calabi-yau manifolds, like this http://en.wikipedia.org/wiki/Calabi-Yau_manifold" (the image on the right). It would appear that http://www.povray.org/" is the appropriate tool (I suspect, after much Googling, that the image was created with POVRay)...
  36. quasar987

    Integration on chains in Spivak's calculus on manifolds

    I would like to discuss this chapter with someone who has read the book. From looking at other books, I realize that Spivak does things a little differently. He seems to be putting less structure on his chains (for instance, no mention of orientation, no 1-1 requirement and so on), and as a...
  37. K

    Check my work (Spivak problem in Calculus on Manifolds)

    Problem: given compact set C and open set U with C \subsetU, show there is a compact set D \subset U with C \subset interior of D. My thinking: Since C is compact it is closed, and U-C is open. Since U is an open cover of C there is a finite collection D of finite open subsets of U that...
  38. K

    Understanding Spivak's "Calculus on Manifolds" - Ken Cohen's Confusion

    Working through Spivak "Calculus on Manifolds." On p. 7, he explains that "the interior of any set A is open, and the same is true for the exterior of A, which is, in fact, the interior of R\overline{}n-A." Later, he says "Clearly no finite number of the open sets in O wil cover R or, for...
  39. A

    Typo in spivak's calculus on manifolds?

    In the first problem set of chapter 1, problem 1-8(b) deals with angle preserving transformations. In the newest edition of the book the problem is stated If there is a basis x_1, x_2, ..., x_n and numbers a_1, a_2, ..., a_n such that Tx_i = a_i x_i, then the transformation T is angle...
  40. Fra

    Emergent statistical manifolds

    Does anyone know of any research programs out there that are considering physical structure formation in terms of emergent, self-organizing statistical manifolds, and does so without starting from some reasonable first principles without any structure or preconception of manifold, or ad hoc...
  41. MathematicalPhysicist

    What are the main considerations when defining a smooth manifold?

    In the attached pdf file i have a few questions on manifolds, I hope you can be of aid. I need help on question 1,2,6,7. here's what I think of them: 1. a) the definition of a smooth manifold is that for every point in M we may find a neighbourhood W in R^k of it which the intersection W...
  42. M

    Definition of arc length on manifolds without parametrization

    Curves are functions from an interval of the real numbers to a differentiable manifold. Given a metric on the manifold, arc length is a property of the image of the curves, not of the curves itself. In other word, it is independent of the parametrization of the curve. In the case of the...
  43. Cincinnatus

    Hilbert Manifolds: An Infinite Dimensional Analogy to Smooth Manifolds?

    Can you define a space that is locally homeomorphic to an infinite dimensional Hilbert space analogously to how you define a (smooth) manifold by an atlas defining local homeomorphisms to R^n? So the charts would just map to the Hilbert space rather than R^n. Could the rest of the definition...
  44. E

    Dimentional reduction: branworld or Yau-calibi manifolds?

    string theorists, there are two approaches to reducing 11 dimensions to 4, they are large and we are stuck on one, or they are compactified, too small to see. Which approach makes the most contact with physics? Is it possible to have 1-2 large dimensions and a 4-folded Yau-Calbi space...
  45. I

    How Does a Function on Manifolds Change When Transferred from M to N?

    Assume you have two manifolds M and N diffeomorphic to another. Also, there is a real-valued function f defined on M. What happens with f when you go from M to N? How is f related to N? thanks
  46. K

    Riemannian Manifolds and Completeness

    Homework Statement Suppose that for every smooth Riemannian metric on a manifold M, M is complete. Show that M is compact. 2. The attempt at a solution I'm honestly not too sure how to start this question. If we could show that the manifold is totally bounded we would be done, but I'm...
  47. G

    Manifolds / Lie Groups - confusing notation

    Hi there, I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is...
  48. G

    Manifolds / Lie Groups - confusing notation

    I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is what I've...
  49. E

    Spivak's Calculus on Manifolds problem (I). Integration.

    Homework Statement If A\subset\mathbb{R}^{n} is a rectangle show tath C\subset A is Jordan-measurable iff \forall\epsilon>0,\, \exists P (with P a partition of A) such that \sum_{S\in S_{1}}v(S)-\sum_{S\in S_{2}}v(S)<\epsilon for S_{1} the collection of all subrectangles S induced by P such...
  50. W

    Kahler Manifolds: Understanding Mutual Compatibility

    Hi, everyone: I am doing some reading on the Frolicher Spec Seq. and I am trying to understand better the Kahler mflds. Specifically: What is meant by the fact that the complex structure, symplectic structure and Riemannian structure (from being a C^oo mfld.) are "mutually...
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