Differential complexes for the Grushin distributions

Transcription

1 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 1/16 Differential complexes for the Grushin distributions Michael Eastwood [ Ovidiu Calin Der-Chen Chang ] Jan Slovák Vladimír Souček Australian National University

2 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 2/16 Grushin distributions Standard coördinates (x,y) on R 2 k th Grushin fields Y x k y bracket generating NB Xf = 0,Y f = 0 f locally constant. Cf + ix y Mizohata Coördinates (x,y,t) on R 3 Y t + x2 y + i y + (ix y) t Martinet (Cf Darboux) Lewy (Cauchy-Riemann-Heisenberg)

3 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 3/16 Zeroth Grushin de Rham complex Standard coördinates (x,y) on R 2 Y y NB Xf = 0,Y f = 0 f locally constant. Xf = g Y f = h } Y g Xh = 0 On any smooth 2-manifold the de Rham complex. 0 R Λ 0 d Λ 1 d Λ 2 0, locally the converse is true

4 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 4/16 The Rumin complex Coördinates (x,y,t) on R 3 Y t + x y Z [X,Y ] = y NB Xf = 0,Y f = 0 f locally constant. Xf = g Y f = h } { XY g X 2 h + Zg = 0 Y Xh Y 2 g Zh = 0 locally the converse is true On any contact 3-manifold 0 R Λ 0 d H Λ 1 d (2) H H Λ 1 H L d H Λ 3 0, the Rumin complex.

5 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 5/16 Rumin cont d Contact 3-manifold: H TM such that [H,H] = TM. 0 L Λ 1 Λ 1 H 0 bracket generating 0 R Λ 0 d Λ 1 d Λ 2 d Λ 3 0 Λ 0 Λ 1 H Λ 2 H + + L Λ 1 H L Λ2 H L Levi second order!

6 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 6/16 (2,3,5)-geometries Traditional coördinates (x,y,p,q,z) on R 5 + p y + q p + q2 z Y q generate a 2-plane distribution H in R 5 and then and then Z [X,Y ] = p 2q z W [X,Z] = y U [Y,Z] = 2 z (and then everything else commutes).

7 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 7/16 (2,3,5) cont d H TM H [H,H] [H, [H,H]] = TM }{{} rank 2 rank 3 rank 5 bracket generating and more besides NB Frame X,Y Γ(H). Xf = 0,Y f = 0 f locally constant. 0 R Λ 0 Λ 1 H??? aweeyyfied For any (2, 3, 5)-geometry st rd nd rd 1 1 st 5 0 BGG... R Bryant, ME, R Gover, K Neusser

8 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 8/16 First Grushin Standard coördinates (x,y) on R 2 Y x y Z [X,Y ] = y NB Xf = 0,Y f = 0 f locally constant. Xf = g Y f = h } { XY g X 2 h + Zg = 0 Y Xh Y 2 g Zh = 0 conversely? Locally (on U open R 2 contractible) Xf = g } { XY g X 2 h + Zg = 0 Y f = h + C Y Xh Y 2 g Zh = 0 constant

9 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 9/16 First Grushin versus contact versus Y x y Y t + x y Z [X,Y ] = y NB 0 E 2 E 3 / t E 3 0 Symmetry reduction Z [X,Y ] = y on R 2 on R 3 / t is a symmetry t t t t Diagram chasing

10 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 10/16 First Grushin complex 0 E 2 3 f 2 4 Xf Y f 3 5 E 2 E g h Y x y X2 h (XY + Z)g Y 2 g (Y X Z)h Z [X,Y ] = y E 2 E E p q 3 5 Xq + Y p here and here else exact Local cohomology is R O Calin, D-C Chang, ME

11 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 11/16 Second Grushin Y x 2 y versus Engel on R 4 Z [X,Y ] = 2x y W [X,Z] = 2 y X = Z [X,Y ] = t + 2x y Y = s + x t + x2 y W [X,Z] = 2 y BGG... R Bryant, ME, R Gover, K Neusser (B Doubrov) 0 0 ր ց րց րց ց ր 2 2

12 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 12/16 Second Grushin complex On R 4 Λ 0 Λ 1 Λ 2 Λ 3 Λ ր 2 nd 1 st ց ր ր + ր ր rd rd ց ր 2 1 st reduce using 1 0 s and t 2 nd O Calin, D-C Chang, ME arxiv:

13 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 13/16 Third Grushin complex X = Y = x 3 Z [X,Y ] W [X,Z] U [X,W] y Look for symmetry reduction from R 5 X = Y = r + x s + x2 t + x3 y Cohomology? Ranks=1, 2, 3, 3, 2, 1? Task for babies?

14 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 14/16 Task for babies? Yes! Everything works as expected a complex 2 nd st 2 ր ց 4 th 1 2 րց Integrability conditions Xf = g Y f = h } 2 1 ց ր reduce using r s t { Y 2 g = Y Xh Zh X 4 h = X 3 Y g + X 2 Zg + XWg + Ug XY Zg 2Y Wg + Z 2 g = X 2 Zh 3XWh + 2Uh Remark On R 2 also 2Y Zg = 2XZh 5Wh Conundrum: who ordered that? ME, J Slovák, V Souček

15 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 15/16 Fourth Grushin Not a task for babies! Slow growth distribution (2, 3, 4, 5, 6) SL(2, R) 4 R 2 Not a parabolic geometry: {( )} 0 Cf B Doubrov & A Medvedev EG Lie algebra cohomology ranks are 1, 2, 3, 4, 3, 2, 1 and then 1, 2, 4, 6, 6, 4, 2, 1 for slow growth (2, 3, 4, 5, 6, 7) et cetera

16 Vector Distributions and Related Geometries at the Banach Centre, Warsaw p. 16/16 THE END THANK YOU