Wikipedia : Manifold | History of manifolds and varieties | Orbifold
Videos :
The Mystery of 3-Manifolds (2010)
Bill Thurston (1946-2012)
What is a Manifold? (Paris, 2010) by
Misha Gromov (1943-).
Vector Bundles, Gauge theory (1:33:28) K Raviteja (2015年08月27日).
What is a Manifold? by Bijan Haney (2016)
1: Point-Set Topology.
2: Topological Concepts.
3: Separation.
4: Countability & Continuity.
5: Compactness & Connectedness.
6: Metrizable Manifolds.
7: Differentiable Manifolds.
8: Diffeomorphisms.
9: Tangent Space.
10: Basis Vectors.
11: Cotangent Space.
12: Fiber Bundles.
14: Tangent Bundle.
15: Quotient Spaces.
16: Moebius Strip.
17: Metric Spaces.
18: Homotopy.
A topological manifold of dimension n is a metrizable space (a second-countable Hausdorff space) locally homeomorphic to Rn (or an open ball thereof).
Parametrization, chart, atlas.
Embedding theorem : Every manifold of dimension n can be embedded in R2n+1
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Topological manifold
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Differentiable manifold
Whitney embedding theorem
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Hassler Whitney (1907-1989)
Topological manifold (18:03)
and
smooth manifold (27:09)
by Harish Seshadri (2017年08月29日).
"Differential Topology" (AMS Lectures at Cornell, Aug. 30 - Sept. 2, 1965) by
John W. Milnor (1931-)
1 : Definitions. Mappings.
2 : Poincaré conjecture. Tangent bundles.
3 : Grassmann manifold.
Differentiable Manifolds (1:43:11)
by Frederic P. Schuller (#7, 2016年03月12日).
One important special case is the connected sum of manifolds of identical dimensions, off a given pair of points. This is done by cutting off a small enough manifold homeomorphic to a ball around both points and gluing together the two boundaries so created.
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Connected sums | Disc theorem (1960) | Richard S. Palais (1931-)
A smooth curve g in a smooth manifold M is, by definition, a C¥ function from R (or any interval thereof) to M.
What's traditionally called the tangent vector to g at point g at point p = g at point g(0) is best defined as the linear form (formally called the directional derivative operator along g at point p) which maps any form of f of C¥(M) to the scalar ( f o g)'(0).
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Two curves which have the same tangent vector at a given point are said to be tangent to each other.
Diffeomorphism
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Smoothness classes
Differential structures (1:44:14)
by Frederic P. Schuller (Lec 09, 2015年09月21日).
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Pushforward
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Pullback
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Differential forms
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Partial derivatives
Manifolds, Derivations & Push-Forwards (59:50)
by James Cook (2015年11月09日).
Loosely speaking, a topological manifold is a topological space which is locally homeomorphic to a d-dmensional Euclideam space.
Formally, a paracompact Hausdorff space M is a d-dimensional (topolological) manifold when every point x of M possesses an open neighborhood homeomorphic to a subset of Rd.
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Differential forms
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Partial derivatives
Topological manifolds and manifold bundles (1:49:17)
by Frederic Schuller (Lec 06, 2015年09月21日).
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A compact m-dimensional Riemannian manifold can be smoothly embedded isometrically in a Euclidean space of (3m+11)m/2 dimensions.
Embedding
Difference between immersion and embedding. (StackExchange, 2011年09月28日).
Nash Embedding Theorem (13:41)
by Edward T. Crane (Numberphile, 2015年05月31日)
Embedding a Torus (12:57)
by James Grime (Numberphile, 2015年05月31日)
