differentialgeometrie.txt

=== Vektoren und Kovektoren

Intuition: http://profstewart.org/pm2/ln11.pdf

A 1-form is a function which grows proportionally to how fast you are moving.
Thus it doesn't matter how you parametrize the curve you are moving on - you
either end up integrating a smaller function for a longer period of time, or a
bigger function for a shorter period of time. This is why you can't integrate
functions on manifolds - they have no intrinsic "unit speeds", because there
are many choices of local coordinates - but you can still integrate
differential forms. k-forms just generalize this to higher dimensions. --
Steven Gubkin

David Bachman. A Geometric Approach to Differential Forms.
http://arxiv.org/abs/math/0306194v1

https://dl.dropboxusercontent.com/u/828035/Mathematics/forms.pdf
von sigfpe!

Lillian Rosanoff Lieber: Tensors, the facts of the Universe.
(via Daniel Fleisch: A Student's Guide to Vectors and Tensors)


=== Torsion

http://mathoverflow.net/questions/20493/what-is-torsion-in-differential-geometry-intuitively


=== Schöne Definition von Differentialformen

https://golem.ph.utexas.edu/category/2010/01/quasicoherent_stacks.html#c030914
Definiere auf Testräumen, dann allgemeiner Nonsens.


=== Theorem von Peetre

Jeder "Operator" Gamma^infty(E) --> Gamma^infty(F) (linear auf Schnitten,
Träger vergrößernd) ist ein Differentialoperator.

https://en.wikipedia.org/wiki/Peetre_theorem


=== Integration holomorpher Funktionen

* Das Integral von 1/z dz über den Einkeitskreis ist tau i, nicht Null.
  Das könnte man aber erwarten, wenn man den Einheitskreis auf der riemannschen
  Zahlenkugel zusammenzieht. Augenscheinlich ist das okay, denn 1/z ist bei
  z = infty definiert und holomorph (mit Funktionswert Null). Dabei hat man
  aber nicht beachtet, dass der volle Integrand, die Differentialform 1/z dz,
  bei z = infty einen Pol hat (erster Ordnung).


=== Faserprodukt von Mannigfaltigkeiten

https://mathoverflow.net/questions/296619/existence-of-fiber-product-of-manifolds-with-one-given-morphism

* Faserprodukte längs Submersionen existieren.
* Faserprodukte von transversalen Morphismen existieren.


=== Literatur

* http://www.cs.columbia.edu/~keenan/Projects/DGPDEC/paper.pdf
  Discrete Exterior Calculus
  anwendungsorientiert

* https://people.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf
  Introduction to Differential Topology, Robbin und Salamon.

* https://www.mat.univie.ac.at/~michor/kmsbookh.pdf
  Here we present the Frölicher–Nijenhuis bracket (a natural extension of the
  Lie bracket from vector fields to vector valued differential forms) as one of
  the basic structures of differential geometry, and we base nearly all
  treatment of curvature and Bianchi identities on it. This allows us to
  present the concept of a connection first on general fiber bundles (without
  structure group), with curvature, parallel transport and Bianchi identity,
  and only then add G-equivariance as a further property for principal fiber
  bundles. We think, that in this way the underlying geometric ideas are more
  easily understood by the novice than in the traditional approach, where too
  much structure at the same time is rather confusing.


=== Nächste Schritte

* Verstehen, inwieweit ein garbentheoretischer Zugang hilft.
  Etwa: Schnittzahl sollte chi(O_A tensor O_B) sein.