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Wikipedia :   Manifold   |   History of manifolds and varieties   |   Orbifold

(2014-12-23)   Manifold   (French:  variété)  and  homotopy type.
Topological, Piecewise-linear or Smooth Manifolds.

A  topological manifold  of dimension  n  is a  metrizable space  (a second-countable  Hausdorff space)  locally homeomorphic to  ⁿ  (or an  open ball  thereof).

Parametrization, chart, atlas.

Embedding theorem :   Every manifold of dimension  n  can be  embedded  in  ²ⁿ⁺¹

Topological manifold   |   Differentiable manifold
 
Whitney embedding theorem   |   Hassler Whitney (1907-1989)
 
Topological manifold (18:03)  and  smooth manifold (27:09)  by  Harish Seshadri  (2017-08-29).
 
"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, 2016-03-12).

(2020-10-03)   Direct sum of two manifolds
The dimension of the sum is the sum of the dimensions.

Diffeomorphism   |   Smoothness classes

(2020-10-03)   Connected sum of two manifolds   (fiber sum).
Cut off two identical submanifolds and glue the matching boundaries.

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.

Connected sums   |   Disc theorem (1960)   |   Richard S. Palais (1931-)

(2020-09-15)   Diffeomorphism

Diffeomorphism   |   Smoothness classes

(2020-10-05)   Tangent verctors and  tangent bundle  thereof.

A smooth curve  g  in a smooth manifold    M  is,  by definition,  a  C^¥  function from    (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).

Two curves which have the same tangent vector at a given point are said to be tangent to each other.

Diffeomorphism   |   Smoothness classes
 
Differential structures (1:44:14)  by  Frederic P. Schuller  (Lec 09, 2015-09-21).

(2020-09-27)   Pushforward:  Total derivative  of a smooth map.
The  dual  operation is a  pullback.

Pushforward   |   Pullback   |   Differential forms   |   Partial derivatives
 
Manifolds, Derivations & Push-Forwards (59:50)  by  James Cook  (2015-11-09).

(2020-10-03)   Toppological Manifolds and Bundles.

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  ^d.

Differential forms   |   Partial derivatives
 
Topological manifolds and manifold bundles (1:49:17)  by  Frederic Schuller  (Lec 06, 2015-09-21).

(2020-09-15)   Immersion  of one differentiable manifold into another:
Differentiable map whose derivative is everywhere  injective.

Immersion

(2020-09-15)   Embedding  of one differentiable manifold into another:
An embedding is an  injective  immersion  (i.e., no "self-intersections").

Nash Embedding Theorem :

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, 2011-09-28).
 
Nash Embedding Theorem (13:41)  by  Edward T. Crane  (Numberphile, 2015-05-31)
Embedding a Torus (12:57)  by  James Grime  (Numberphile, 2015-05-31)