# Seminar on the h-principle

Prof. Bernd Ammann,

# Zoom access

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## Content of the Seminar

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=S^{2} and N=ℝ^{3} one may conlude that in the class of immersions S^{2} →ℝ^{3} 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

## Requirements

Solid knowledge about differential geometry.
## Time and Place

Tuesday, 16:15 (or 16:30) to 18:00.
## Program

The program of the seminar.
## Registration

Please register via Email to Bernd Ammann.
## Related web pages

Bernd Ammann, 05.06.2021

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