Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg

Hinweis Bitte informieren Sie sich auf den jeweiligen GRIPS-Seiten über den digitalen Ablauf der Lehrveranstaltungen.

English Note For our digital courses all relevant information can be found on the appropriate GRIPS sites.

Seminar: The h-principle
Semester
SoSe 2021

Lecturer
Bernd Ammann

Type of course (Veranstaltungsart)
Seminar

Contents
The h-principle is an extremely powerful machinery to construct geometric objects with certain properties. There is a very broad field of applications.

On the technical side a major notion is the notion of "partial differential relation", which should be understood as "some" condition on partial derivatives on a function between manifolds. For fixed manifolds M and N examples of such relations are the following

  • f:M → N is an immersion,
  • g is a positively curved Riemannian metric on M,
  • similar for negatively curved Riemannian metrics,
  • g is a Riemannian metric with sectional curvature between 0.999 and 1 on M.
For all these examples, and for many more ones, the h-principle provides solutions, and often it also allows to determine the topology of the space of all solutions.

On important application is the Smale-Hirsch theorem: Let M and N be manifolds such that M is non-compact and connected or such that M is of lower dimension than N. Then the map

d:Imm(M,N)→Mon(TM,TN)
is a weak homotopy eqiuvalence. Here Imm(M,N) denotes the space of all Immersions from M to N, d denotes the differential and Mon(TM,TN) denotes the space of vector bundle monomorphisms. In the special case M=S2 and N=ℝ3 one may conlude that in the class of immersions S2 →ℝ3 there is a path from the standard immersion x↦ x to its negative x↦ -x, a so-called sphere eversion, which maybe interpreted as turning the inside of the sphere out. For a movie see here for some preliminary facts and here (after 5 minutes) for a movie visualizing such an eversion. See here for an alternative link.

Other applications are: any connected, non-compact manifold manifolds admits a metric with sectional curvature between 0.999999 and 1 and other ones between -1 and -0.999999999. Further applications range into contact and symplectic geoemtry.

If time admits, we will also discuss "convex integration" which is a variant of these methods. This method allows, for example, the construction of metrics with negative Ricci-curvature of arbitrary closed manifolds of dimension at least 3.

Literature

  • Y. Eliashberg, N. Mishachev, Introduction to the h-principle, Graduate Studies in Mathematics 48, AMS
  • H. Geiges, h-principle and flexibility in geometry, Memoirs AMS 164 (2003), no. 779
  • M. Gromov, Partial differential relations, Springer


Recommended previous knowledge
Solid knowledge about differential geometry

Time/Date
Tuesday 16-18

Location
via zoom

Course homepage
http://www.mathematik.uni-regensburg.de/ammann/lehre/2021s_hprinciple
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration
  • To register, please send an email to Bernd Ammann
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Presentation: Giving a seminar talk of roughly 90 minutes
Examination (Prüfungsleistungen)
  • Detailed written report of the seminar talk
Regelungen bei Studienbeginn vor WS 2015 / 16
  • Benotet:
    • O. g. Studienleistung und o. g. Prüfungsleistung; die Note ergibt sich aus dem Seminarvortrag
  • Unbenotet:
    • O. g. Studienleistung
Modules
MV, MSem

ECTS
Siehe Modulkatalog. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn
vor WS 15/16