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. D

    Understanding the notion of a tangent bundle

    I've been reading up on the definition of a tangent bundle, partially with an aim of gaining a deeper understanding of the formulation of Lagrangian mechanics, and there are a few things that I'm a little unclear about. From what I've read the tangent bundle is defined as the disjoint union of...
  2. caffeinemachine

    MHB To Prove that The Level Set Of A Constant Rank Map is a Manifold

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  3. C

    Bundle and differential manifold

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  4. D

    Vectors in Tangent Space to a Manifold Independent of Coordinate Chart

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  5. D

    Differentiability of a function on a manifold

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  6. binbagsss

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  7. W

    If the Product Manifold MxN is orientable, so are M,N.

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  8. E

    Is a Subset of the Eigenvector Matrix in PCA Equivalent to a Submanifold?

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  9. V

    Big Bang: A True Singularity That is Coordinate Independent

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  10. W

    Hyperbolic Manifold With Geodesic Boundary?

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  11. Spinnor

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  12. Spinnor

    What can a complex manifold do for me that real manifolds can't.

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  13. Spinnor

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  14. Spinnor

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  15. V

    One-forms in differentiable manifolds and differentials in calculus

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  16. V

    Find Null Paths in Differentiable Manifolds Using One-Forms

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  17. camilus

    The Grassmanian manifold's topology

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  18. A

    Push Forward on a Product Manifold.

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  19. K

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  20. shounakbhatta

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  21. I

    What is a Manifold? Curvature Explained in 3D Space

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  22. W

    Differential Form on Product Manifold

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  23. Mandelbroth

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  24. ChrisVer

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  25. B

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  26. C

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  28. C

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  29. C

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  30. N

    Is GL(n,C) the biggest complex manifold within GL(2n,R)?

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  31. K

    Closed ball is manifold with boundary

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  32. benzun_1999

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  33. C

    Are left-invariant fields mapped onto the manifold Killing vectors?

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  34. S

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  35. C

    Example of a topological manifold without smooth transition functions.

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  36. P

    Deductions about extending a pseudo-riemannian manifold

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  37. N

    In how many ways can you embed a Riemannian manifold in R^n?

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  38. P

    How to determine extension of a pseudo-Riemannian manifold

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  39. C

    Idea behind topological manifold definition.

    The usual definition of an n-dimensional topological manifold M is a topological space which is 'locally Euclidean', by which we mean that: (1) every point in M is contained in an open set which is homeomorphic to ##\mathbb{R}^n##. (2) M is second countable. (3) M is an Hausdorff space...
  40. W

    Orientation of a Submanifold, given an Orientation of Ambient Manifold

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  41. C

    Is the Metric g a Complex Manifold?

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  42. M

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  43. S

    Does a compact manifold always have bounded sectional curvature?

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  44. mnb96

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  45. mnb96

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  46. B

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  47. Philosophaie

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  48. TrickyDicky

    Are singularities part of the manifold?

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  49. F

    Is General Relativity Dependent on the Existence of a Manifold?

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  50. Mandelbroth

    Plausibility of a Discrete Point Manifold

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