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Rigidity ppt
- 1. On the rigidity of geometric structures and its connections to algebra and topology. T.Venkatesh School of Mathematics and computing Science R.C.U, Belagavi J.V.Ramana Raju School of Graduate Studies Jain University Bangalore
- 2. SLIDE-2 About a decade ago the second author happened to listen to a talk on “Geometric Rigidity” delivered by an algebraist… and the results seemed to be couched in the language of algebra and representation theory. The effort here is to understand rigidity results from a geometric viewpoint.
- 3. SLIDE-3 • “Rigidity” is not defined precisely and so it covers various situations where a weaker condition on a topological space forces a stronger condition to hold. •Colloquially the term Rigid refers to “ being stubborn when subjected to change” •We shall see situations where such a property holds, and situations where there is a lack of such rigidity. All manifolds considered here are compact without boundary.
- 4. SLIDE-4 Suppose X1 , X2 are two closed genus ‘g’ surfaces, then we know that X1~X2, where the equivalence is a homeomorphism and also a diffeomorphism. We refer to this property as topological rigidity. Now one can ask the question as to how many non-isometric classes of such 2- manifolds exist?. In other words how big is the deformation space ?
- 5. SLIDE-5 The answer to this question is understood via Teichmuller theory, that for a genus g surface (g≥2) one can continuously deform the space to obtain parameter space of dimension 6g-6. We refer to this situation as the lack of geometric rigidity.
- 6. SLIDE-6 Whenever X1 , X2 are two manifolds of the same homotopy type, if they are necessarily isometric, then we say that X1 is geometrically rigid Remark: For genus 2 surfaces one can explicitly show the construction of non- isometric classes.
- 7. SLIDE-7 However in dimension 3 and above, in case of hyperbolic manifolds one cannot deform in the way we did in the surfaces case. Theorem(Mostow,Prasad) Suppose M,N are two hyperbolic manifolds of finite volume. If M and N have isomorphic fundamental groups, then M and N are necessarily isometric.
- 8. SLIDE-8 •In other words geometric invariants like hyperbolic length , volume etc are all topological invariants. •Thus we can say that the geometric structure of a hyperbolic 3-manifold is extremely rigid •Remark: In the case of flat Riemannian manifolds, Bieberbach theorem is a result
- 9. SLIDE-9 MORE HISTORY It was M.Berger who gave the first Rigidity result from a differential geometric viewpoint. These results fit into what may be called as “curvature rigidity” Theorem: If M is a manifold such that all its sectional curvatures lie between 1 and 4 or between -4 and -1, then M is either a sphere or is isometric to a rank-1 symmetric space
- 10. SLIDE-10 MORE HISTORY…. There is an extensive body of work by Farrell and Jones for the case of non- positively curved manifolds of dimension atleast 5. Richard Schoen etal have charecterised spherical spaceforms
- 11. SLIDE-11 Theorem (Schoen): Suppose M is a compact 2-manifold with scalar curvature positive everywhere. Then M is either S2 or RP2 e Similar result holds for all even dimensional compact manifolds. A result of Gromov and Lawson says that the torus is rigid in the following sense:
- 12. SLIDE-12 Theorem (Gromov, Lawson): Let g be a Riemannian metric of non-negative scalar curvature on the torus Tn Then g must be necessarily flat. Applying this to the two torus we can say that the 2-torus exhibits curvature rigidity.
- 13. SLIDE-13 Algebraic versions: The Rigidity phenomena in the sense of Mostow has several algebraic versions. The lack of rigidity in the case of surfaces can be attributed to the presence of families of non-conjugate lattices in SL2(R) [by the theory of uniformization]