Gray's stability theorem

The stability set of Gray is a fundamental mathematical theorem in the field of contact geometry . It says that contact structures can only be deformed by isotopias of the entire manifold, that is, the module space of the contact structures is a discrete space .

Gray's theorem

Be with a smooth family of contact structures on a closed manifold . Then there is an isotopy of such that for all of the diffeomorphism ${\displaystyle \xi _{s}}$ ${\displaystyle s\in \left[0,1\right]}$ ${\displaystyle M}$ ${\displaystyle (\psi _{s})_{s\in \left[0,1\right]}}$ ${\displaystyle M}$ ${\displaystyle s\in \left[0,1\right]}$

${\displaystyle \psi _{s}\colon (M,\xi _{0})\to (M,\xi _{s})}$

is a contactomorphism , d. i.e. it applies to everyone . ${\displaystyle (\psi _{s})_{*}\xi _{0}=\xi _{s}}$ ${\displaystyle s\in \left[0,1\right]}$

literature

• John W. Gray : Some global properties of contact structures. Ann. of Math. (2) 69 1959 421-450.
• Hansjörg Geiges : An introduction to contact topology. Cambridge Studies in Advanced Mathematics, 109. Cambridge University Press, Cambridge, 2008. ISBN 978-0-521-86585-2

Individual evidence

1. ^ Theorem 2.2.2 in Geiges, op.cit.