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.

View More On Wikipedia.org
Recent contents Top users
  1. W

    Highdimensional manifold reconstruction

    Suppose I have a highdimensional space \mathbb{R}^N that is sparesely populated by a finite set of samples \{ \mathbf{x} \}_{1 \le i \le k} , for example N = 500, k = 100. I assume the points x to be sampled from a n-manifold embedded in \mathbb{R}^N, where n << N. From a mathematical point of...
  2. L

    What is a Differential Structure on a Manifold?

    Hi, I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help! Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for...
  3. J

    Definition of Manifold

    I have a question about the definition of a manifold given in my analysis book. Here is the definition: Let 0 < k \le n . A k-manifold in \mathbb{R}^n of class C^r is a set M \subset \mathbb{R}^n having the following property: For each p in M, there is an open set V of M containing p...
  4. I

    Foliation of 4-dimensional connected Hausdorff orientable paracompact manifold

    Hi all! I am reading a book on Classical Electrodynamics (Hehl and Obukhov, Foundations of Classical Electrodynamics, Birkhauser, 2003). In this manual I found the following statement: ============ Consider a 4-dimensional differentiable manifold which is: -connected (every 2 points are...
  5. H

    Concerning charts on a manifold

    I'm new to manifolds, so please forgive me if this sounds ignorant. I was just wondering whether the charts of a smooth manifold (within some atlas) always "overlap". If I'm not mistaken they map to open subsets of R^n, and being homeomorphisms should have the inverse image as open. But I'm not...
  6. A

    Differential manifold without connection: Is it possible?

    We all are familiar with the kind of differential geometry where some affine connection always exists to relate various tangent spaces distributed over the manifold, and from this connection two fundamental tensors, namely the Cartan's torsion and the Riemann-Christoffel curvature, arise. Is it...
  7. W

    Is the Transversal Intersection of Manifolds a Manifold?

    Hi, All: Given manifolds M,N (both embedded in $R^n$, intersecting each other transversally, so that their intersection has dimension >=1 ( i.e. n -(Dim(M)-Dim(N)>1) is the intersection a manifold? Thanks.
  8. J

    What is the definition of integration over a manifold and what does it measure?

    When we integrate a scalar map over a manifold M, what exactly are we measuring? If M is the unit circle in R^2, then regular Riemann integration of the function f = 1 over it will yield the volume of a cylinder of height 1. Okay, no problem. Now, if we integrate f = 1 over the unit circle in...
  9. G

    What metric from given manifold?

    Given a manifold as algebraic variety, say sphere, how do we obtain possible metrics? how do we classify them? If spcaetime manifold is n-sphere, Einstein's vacuum (for now) equation would be some special metric among many other possible metrics? i'm curious what role Einstein's equation...
  10. TrickyDicky

    Optimal Number of Charts for Smooth Manifolds in Dimension 2

    What is the least number of charts needed to specify a given smooth manifold, for simplicity of dimension 2? For instance the minimum number of charts to cover a torus, or a 2-sphere or a 2D 1-sheet hyperboloid? I would think it goes with the definition of manifold that in any case you need at...
  11. Y

    Convergence of a sequence of points on a manifold

    I have a question regarding the following definition of convergence on manifold: Let M be a manifold with atlas A. A sequence of points \{x_i \in M\} converges to x\in M if there exists a chart (U_i,\phi_i) with an integer N such that x\in U_i and for all k>N,x_i\in U_i \phi_i(x_k)_{k>N}...
  12. alemsalem

    Is a (smooth) manifold allowed to have different dimensions in different points.

    obviously in one coordinate neighborhood it can't.. I'm thinking of a line which smoothly develops into a surface : -----<< what particular properties would this object have.. Thanks :)
  13. D

    Notation for basis of tangent space of manifold

    I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it? for example, i know...
  14. J

    Orientability of 1-Dimensional Manifolds: A Closer Look

    I have the result that any 1 dim topological manifold is either R or S1. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices. i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the...
  15. B

    Proving g is a One-Param Group of Diffeomorphisms on a Manifold

    Homework Statement Let M be a differentiable manifold and g: \Re \times M \rightarrow M, (t,x) \rightleftharpoons g^{t}x be a map such that the following conditions are satisfied. i) g is a differentiable map. ii) The map \Re \rightarrow Aut(M), t \rightleftharpoons g^{t} is a...
  16. T

    Exploring the Geometric Connection between Tangent Spaces and Rn

    Hi everyone, On the Wikipedia page for Tangent space there is a definition of the tangent space at a point x using equivalence classes of curves. It mentions that the tangent space TxM is in bijective correspondence with Rn. My first question is simply: is there an easy geometric way using...
  17. L

    Manifold / Atlas / Chart (Building Simple Example)

    I’m studying GR and am curious about manifold, atlas and charts. I have an idea for building a simple example, in one dimension, and wanted to ask if what I’m doing below is “legal”/correct. Imagine a space flight that can be divided into three segments: A-B: velocity starts at zero and...
  18. S

    Structuring the graph of |x| so it is not a smooth manifold

    Hello, I am learning about smooth manifolds through Lee's text. One thing that I have been pondering is describing manifolds such as |x| which are extremely well behaved but not smooth in an ordinary setting. It is simple to put a smooth structure on this manifold, however that is...
  19. R

    Lorentz Invariance as local limit of Bigger Manifold

    Is it possible that Lorentz invariance is just a lower limit of a larger manifold that has a priveleged frame? Even if Bell's experiments can't transmit signal faster than light. The spirit of relativity is still violated by say instantaneous correlation between 10 billion light years. As...
  20. W

    Length of a curve on a manifold

    Can anyone help with finding the length of a circle (theta) =pi/2 (latitude 90') on the unit sphere. I know it is related to the equation L= integral from 0 to T of Sqrt(g_ij (c't,c't)) The formula is on the wikipedia page called Riemannian manifold so you can get a better idea what it...
  21. W

    Riemannian Manifold: Integral Formula Explained

    Can someone please explain to me how this formula integral from 0 to T of sqrt(g_ij c'(t) c'(t)) I have seen it on wikipedia but don't know how to actually implement the formula.
  22. S

    Spivaks Calculus on a Manifold

    Homework Statement Prove the triangle inequality theorem \leftX-Y\right\rfloor\lfloor\leq\leftZ-Y\right\rfloor\lfloor+\leftY-X\right\rfloor\lfloor. (my computer just shows the latex as a bunch of script so i don't know if that came out right. Homework Equations The Attempt at a...
  23. M

    Compactness of Tangent Bundle: Manifold M

    hello friends :smile: I have a question about the compactness of the tangent bundle: assume that the manifold M is compact, does it make necessarily TM compact ? if not TM, a submanifold of TM (precisely a submanifold of vector norm equal to 1) can be compact?
  24. atomqwerty

    Tangent Space and Manifold of a Cubic Surface

    Homework Statement In which points the surface \{\left(x,y,z\right)\in\Re^{3}|x^{3}-y^{3}+xyz-xy=0\right\} is a differentiable manifold (subvariedad diferenciable in spanish). Calculate its tangent space in the point (1,1,1). Homework Equations NA The Attempt at a Solution I've...
  25. radou

    Is Every Manifold Regular? A Proof Using the Hausdorff Condition

    Homework Statement As the title suggests, I need to show that every manifold is regular. There's probably something wrong with my proof, since I didn't use the Hausdorff condition, and the book almost explicitly states to do so. The Attempt at a Solution So, a m-manifold is a...
  26. M

    Does a Compact Manifold Imply a Compact Tangent Bundle?

    hello friends my question is: if we have M a compact manifold, do we have there necessarily TM compact ? thnx .
  27. I

    Does space-time form a closed manifold around a black hole?

    Mass can curve space-time. Is it possible that space-time around a black hole is so badly curved that it forms a closed 4D manifold?
  28. I

    Non-linear dynamics approach to a manifold of a saddle point using power series

    Homework Statement Im taking a dynamics course and I am using The strogatz book Non-linear Dynamics and Chaos I need to solve a problem that is similar to problem 6.1.14 Basically it consist in the following You have a saddle node at (Ts,Zs) which is (1,1). Consider curves passing through...
  29. R

    Is every one-dimensional manifold orientable?

    Is there any non-orientable one-dimensional manifold ? If not, how to prove it? Thanks!
  30. M

    Let M be a manifold and g a metric over M

    let M be a manifold and g a metric over M . is it true that every subbundle from M must have the same metric g ?
  31. H

    Lie Algebra differentiable manifold

    Okey, I have problem with the foundation of lie algebra. This is my understanding: We have a lie group which is a differentiable manifold. This lie group can for example be SO(2), etc. Then we have the Lie algebra which is a vectorspace with the lie bracket defined on it: [. , .]. This...
  32. G

    Pressure evolution in an engine intake manifold (thermodynamics)

    Hi, I am using a model that estimates pressure (P) in an intake manifold. I think there is a mistake in its equations but I cannot find it. To simplify the problem we can make the following assumptions: - Only air fills the manifold: air comes into the manifold through the throttle (mass...
  33. E

    Immersion and Manifold Question

    Homework Statement Let's assume that M is a compact n-dimensional manifold, then from Whitney's Immersion Theorem, we know that there's an immersion, f: M -> R_2n, and let's define f*: TM --> R_2n such that f* sends (p, v) to df_x (v). Since f is an immersion, it's clear that f* must be...
  34. Phrak

    Understanding Dual Manifolds in General Relativity

    Background. We define vectors in general relativity as the differential operators \frac{\cdot}{d\lambda}=\frac{dx^\mu}{d\lambda}\frac{\cdot}{\partial x^\mu} which act on infinitessimals--dual vectors, df=\frac{\partial f}{\partial x^\mu} dx^\mu \ , as linear maps to reals. However, both...
  35. J

    Integrating a Scalar Map over a Compact Manifold: What's So Special About It?

    First let me write out the definition of a manifold given in my book: Let k > 0 . A k-manifold in \mathbb{R}^n of class C^r is a subspace M of \mathbb{R}^n having the following property: For each p \in M , there is an open set V \subset M containing p , a set U that is open in...
  36. H

    Smooth manifold and constant map

    Suppose M and N are smooth manifold with M connected, and F:M->N is a smooth map and its pushforward is zero map for each p in M. Show that F is a constant map. I just remember from topology, the only continuous functions from connected space to {0,1} are constant functions. With this be...
  37. bcrowell

    Defining a manifold without reference to the reals

    Most definitions I've seen for a manifold are based on the idea that small neighborhoods are homeomorphic to \mathbb{R}^n. To me this feels a little like defining a bicycle as a car that's missing the engine and both the wheels on one side. The real number system is this big, sophisticated piece...
  38. quasar987

    Is every manifold triangulable?

    In Lee's Intro to topological manifolds, p.105, it is written that every manifold of dimension 3 or below is triangulable. But in dimension 4, threre are known examples of non triangulable manifolds. In dimensions greater than four, the answer is unknown. But in Bott-Tu p.190, it is written...
  39. H

    A question about vector space manifold

    If k is an integer between 0 and min(m,n),show that the set of mxn matrices whose rank is at least k is an open submanifold of M(mxn, R).Show that this is not true if "at least k"is replaced by "equal to k." For this problem, I don't understand why the statement is not true if we replace "at...
  40. E

    Projection of a differentiable manifold onto a plane

    For a game I am thinking about making I would need to know how to project points from a differentiable bounded 3-manifold to a Euclidean plane (the computer screen). The manifold would be made from a 3-dimensional space with two balls cut out of it and a hypercylinder glued onto it at the holes...
  41. bcrowell

    Things that can't happen on a manifold

    In a different thread, JesseM raised ( https://www.physicsforums.com/showpost.php?p=2858281&postcount=37 ) what I thought was an interesting question: can a manifold (over the real numbers) contain points that are infinitely far apart? Since a bare manifold doesn't come equipped with a metric...
  42. D

    Pressure drop in a water manifold

    Hi all, This is what I currently have. Water chiller with a flowrate of 2GPM (Choked to 2GPM) Ti= 50F H20 Pi=55-60psi It is a closed loop system. I am looking to split 1 line at 2 GPM (55-60psi) into 8 smaller lines that will run into my cooling jacket. All 8 lines are same length...
  43. P

    Topology on manifold and metric

    Is there any relation between topology on manifold (which comes from \mathbb{R}^n) and topology induced form metric in case of Remanian manifold. What if we consider pseudoremaninan manifold.
  44. G

    Hausdorff condition in differential manifold definition

    Dear all, why is it needed in the diff manifold definition that the base set M is topologically Hausdorf ? Since M is locally homeomorphic with Rn as metric space is Hausdorf, shouldn't this condition be automatically satisfied? Thanks. Goldbeetle
  45. W

    Why the basis of the tangent space of a manifold is some partials?

    it is quite peculiar i know you do not want to embed the manifold into a R^n Euclidean space but still it is too peculiar it is hard to develop some intuition
  46. 0

    Bending of manifold and coordinate change

    I am new to this subject of topology. I want to know if bending and stretching of a manifold is same as a general transformation of a coordinate system drawn on the manifold. Or the mathematical definition of bending and stretching shall equally help.
  47. TrickyDicky

    Doubt about our spacetime manifold

    I understand that accordingt to GR mass curves the spacetime (I'm not referring to spatial curvature k), so that the universe globally considered is a manifold with constant curvature, is this right? If so, is this curvature positive or negative in the current cosmological model?
  48. P

    Weyl Tensor Components in n-Dim Manifold: N=3?

    I found the formula for the number of independent components of Weyl tensor in n-dimensional manifold: (N+1)N/2 - \binom{n}{4} - n(n+1)/2~~~~~N=(n-1)n/2 This expression implies that in 3 dimension Weyl tensor has 0 independent components, so it's 0. Does it implies that any three-dimensional...
  49. E

    Lie algebra of the diffeomorphism group of a manifold.

    I have seen it mentioned in various places that the Lie algebra of the diffeomorphism group of a manifold M is identifiable with the Lie algebra of all vector fields on M, but I have not found a demonstration of this. I can show that the map \rho: Lie(Diff(M)) \to Vect(M), ~~~ \rho(X)_p =...
  50. P

    Two dimensional manifold are conformally flat

    Does anyone know why every 2D manifold is conformally flat.
Back
Top