BI-PARACONTACT STRUCTURES AND LEGENDRE ...paracontact struct. and φ3,ξ,η) is normal as almost...

XIX FALL WORKSHOP “GEOMETRY AND PHYSICS”

Beniamino Cappelletti Montano

CMUC, University of Coimbra (visitor)

BI-PARACONTACT STRUCTURES AND LEGENDRE FOLIATIONS

OPORTO - SEPTEMBER 6, 2010

A contact manifold is a smooth manifold M endowed with a 1-form η such that η ∧ (dη)n ≠ 0 (1) everywhere on M.

Contact manifolds


M is necessarily of odd dimension, 2n+1

Contact manifolds


M is necessarily of odd dimension, 2n+1

There exists a unique vector field ξ, called Reeb vector field, such that

η(ξ) = 1 and dη(ξ,·) = 0.

Contact manifolds

D:=ker(η) is called the contact distribution.

Foliation theory ⇒ D is integrable if and only if η ∧ dη = 0. Thus the geometric interpretation of

η ∧ (dη)n ≠ 0

is that the contact distribution is as far as possible from being integrable.

Contact manifolds

D:=ker(η) is called the contact distribution.

Foliation theory ⇒ D is integrable if and only if η ∧ dη = 0. Thus the geometric interpretation of

η ∧ (dη)n ≠ 0

is that the contact distribution is as far as possible from being integrable. The maximal dimension of an integrable subbundle of D is n.

Contact manifolds

Definition Let (M2n+1,η) be a contact manifold. A foliation F of M2n+1 is called Legendre foliation if

dim(F) = n

T(F) is a subbundle of the contact distribution D

Legendre foliations

More in general, a Legendre distribution is an n-dimensional sub-bundle L of the contact distribution such that

dη(X,X’) = 0

for all X,X’∈Г(L). Note that this condition holds if L is involutive, since

2dη(X,X’) = X(η(X’)) – X’(η(X)) – η([X,X’]).

Legendre foliations

More in general, a Legendre distribution is an n-dimensional sub-bundle L of the contact distribution such that

dη(X,X’) = 0

for all X,X’∈Г(L). Note that this condition holds if L is involutive, since

2dη(X,X’) = X(η(X’)) – X’(η(X)) – η([X,X’]).

M. Y. Pang, The structure of Legendre foliations, Trans. Amer. Math. Soc. 1991

P. Libermann, Legendre foliations on contact manifolds, Differential Geom. Appl. 1990

A. Bejancu, H. Farran, Foliations and geometric structures, Springer, 2006

Legendre foliations

Definition Let (M,η) be a contact manifold.

Almost contact and paracontact structures


An almost contact structure on (M,η) is given by a tensor field φ such that

φ2 = −I + η⊗ξ.



An almost contact structure on (M,η) is given by a tensor field φ such that

φ2 = −I + η⊗ξ.

An almost paracontact structure on (M,η) is given by a tensor field φ such that

� φ2 = I − η⊗ξ

� φ induces an almost paracomplex structure on the contact distribution (i.e. the ±1 eigendistributions of φ|D have equal dimension n).


The geometry of an almost contact / paracontact manifold (M2n+1,φ,ξ,η) is related to almost complex / paracomplex structures by considering

M2n+1 × R and

( )

=

dt

dXηξfXφ

dt

dfXJ ,, m

for all X Γ(TM2n+1) and f C∞(M2n+1×R).



M2n+1 × R and

( )

=

dt

dXηξfXφ

dt

dfXJ ,, m


J2 = −I / J2 = I, respectively.



M2n+1 × R and

( )

=

dt

dXηξfXφ

dt

dfXJ ,, m


J2 = −I / J2 = I, respectively.

When [J,J] ≡ 0,

the almost contact / paracontact manifold M2n+1 is called normal.


It suffices to evaluate [J,J] in the couples of vector fields of type

((X,0),(Y,0)) and ((X,0),(0,d/dt)), for any X,Y Γ(TM2n+1):

[ ] ( ) ( )( )0,,0,, YXJJ

and

[ ] ( )

dt

dXJJ ,0,0,,




[ ] ( ) ( )( ) ( )( ) ( )( )

=

dt

dYXNYXNYXJJ ,,,0,,0,, 21

and

[ ] ( ) ( )( ) ( )( )( )XNXNdt

dXJJ 43 ,,0,0,, =




[ ] ( ) ( )( ) ( )( ) ( )( )

=

dt

dYXNYXNYXJJ ,,,0,,0,, 21

and

[ ] ( ) ( )( ) ( )( )( )XNXNdt

dXJJ 43 ,,0,0,, =

where N(1)(X,Y) := [φ,φ](X,Y) + 2dη(X,Y)ξ

N(2)(X,Y) := ( φXη)(Y) – ( φYη)(X)

N(3)(X) := ( ξφ)X

N(4)(X) := ( ξη)X.


Then

(φ, ξ, η) is normal

N(1) = N(2) = N(3) = N(4) = 0


Then

(φ, ξ, η) is normal

N(1) = N(2) = N(3) = N(4) = 0

N(1)= 0


Definition An almost bi-paracontact structure on a contact manifold (M,η) is given by the triplet (φ1,φ2,φ3), where φ1,φ2,φ3 are tensor fields of type (1,1) on M such that

Bi-paracontact structures


φ12 = φ22 = I – η⊗ξ



φ12 = φ22 = I – η⊗ξ

φ32 = –I + η⊗ξ



φ12 = φ22 = I – η⊗ξ

φ32 = –I + η⊗ξ

φ1φ2 = –φ2φ1 = φ3



φ12 = φ22 = I – η⊗ξ

φ32 = –I + η⊗ξ

φ1φ2 = –φ2φ1 = φ3

For each =1,2 set

Dα+ := {X∈T(M) │ φ X = X} Dα− := {X∈T(M) │ φ X = −X}


Proposition The canonical distributions D1+, D1−, D2+, D2− satisfy the following properties:



1. D1± = {X + φ3X | X D2±}, D2± = {Y + φ3Y | Y D1}.



1. D1± = {X + φ3X | X D2±}, D2± = {Y + φ3Y | Y D1}. 2. For any ≠β,

T(M) = D + ⊕ D − ⊕ Rξ = D ± ⊕ Dβ± ⊕ Rξ.




T(M) = D + ⊕ D − ⊕ Rξ = D ± ⊕ Dβ± ⊕ Rξ.

3. dim(D1+) = dim(D1−) = dim(D2+) = dim(D2−) = n.

It follows that φ1 and φ2 are almost paracontact structures.




T(M) = D + ⊕ D − ⊕ Rξ = D ± ⊕ Dβ± ⊕ Rξ.

3. dim(D1+) = dim(D1−) = dim(D2+) = dim(D2−) = n.

It follows that φ1 and φ2 are almost paracontact structures. Are D + and D − Legendre distributions?


Theorem Let (φ1,φ2,φ3) be an almost bi-paracontact structure on (M,η). Then, for each =1,2,

D + and D − are Legendre distributions

dη is of type (1,1) with respect to φ

N (2) = 0 where

N (2)(X,Y) := ( X

�

η)(Y) – ( Y

�

η)(X).


.


D + and D − are Legendre foliations

N (1) vanishes on the contact distribution where

N (1) := [φ ,φ ] + 2dη⊗ξ.


.


D + and D − are Legendre foliations

N (1) vanishes on the contact distribution where

N (1) := [φ ,φ ] + 2dη⊗ξ.

Definition When, for all =1,2, one among the above two conditions holds, the almost bi-paracontact structure is called integrable.



D + and D − are flat (= ξ is basic) Legendre foliations

N (1) = 0




N (1) = 0

Definition When, for all =1,2, one among the above two conditions holds, the almost bi-paracontact structure is called normal.




N (1) = 0


(φ1,φ2,φ3) is normal ⇔ (φ1,ξ,η), (φ2,ξ,η) are normal as almost paracontact struct. and (φ3,ξ,η) is normal as almost contact struct.




N (1) = 0


(φ1,φ2,φ3) is normal ⇔ (φ1,ξ,η), (φ2,ξ,η) are normal as almost paracontact struct. and (φ3,ξ,η) is normal as almost contact struct.

normality ⇒⇒⇒⇒ integrability


Odd dimension Even dimension

almost bi-paracontact

structure




structure

� Almost quaternion structure of the II kind (Yano, Ako 1973)




structure


� 3-web (Nagy 1988)




structure


� 3-web (Nagy 1988) � Bi-paracomplex structure (Santamaría

1999; Etayo, Santamaría, Trías 2006)




structure



1999; Etayo, Santamaría, Trías 2006) � Anti-hypercomplex structure

(Marchiafava, Nagy 2003)




structure




(Marchiafava, Nagy 2003) � Complex-product structure (Andrada

2005)




structure





2005) � Para-hypercomplex structure (Ivanov,

Zamkovoy 2005)




structure






Zamkovoy 2005)

Obata connection Chern connection

Bi-Lagrangian connection




structure






Zamkovoy 2005)

Obata connection Chern connection

Bi-Lagrangian connection ?


Theorem Let (φ1,φ2,φ3) be an almost bi-paracontact structure on (M,η). For each = 1,2,3 there exists a unique connection � such that


Theorem Let (φ1,φ2,φ3) be an almost bi-paracontact structure on (M,η). For each = 1,2,3 there exists a unique connection � such that i) � ξ = 0



ii) �1φ1 = 0, �1φ2 = η⊗(2h2–h1φ3+φ3h1), �1φ3 = η⊗(2h3–h1φ2+φ2h1)




�2φ1 = η⊗(2h1+h2φ3–φ3h2), �2φ2 = 0, �2φ3 = η⊗(2h3–h2φ1+φ1h2)




�2φ1 = η⊗(2h1+h2φ3–φ3h2), �2φ2 = 0, �2φ3 = η⊗(2h3–h2φ1+φ1h2)

�3φ1 = η⊗(2h1–h3φ2+φ2h3), �3φ2 = η⊗(2h2+h3φ1–φ1h3), �3φ3 = 0



ii) ��1φ1 = 0, �1φ2 = η⊗(2h2–h1φ3+φ3h1), �1φ3 = η⊗(2h3–h1φ2+φ2h1)

�2φ1 = η⊗(2h1+h2φ3–φ3h2), ��2φ2 = 0, �2φ3 = η⊗(2h3–h2φ1+φ1h2)

�3φ1 = η⊗(2h1–h3φ2+φ2h3), �3φ2 = η⊗(2h2+h3φ1–φ1h3), ��3φ3 = 0



ii) ��1φ1 = 0, �1φ2 = η⊗(2h2–h1φ3+φ3h1), �1φ3 = η⊗(2h3–h1φ2+φ2h1)

�2φ1 = η⊗(2h1+h2φ3–φ3h2), ��2φ2 = 0, �2φ3 = η⊗(2h3–h2φ1+φ1h2)

�3φ1 = η⊗(2h1–h3φ2+φ2h3), �3φ2 = η⊗(2h2+h3φ1–φ1h3), ��3φ3 = 0

iii) T (φ X,Y) – T (X,φ Y) = 2(dη(φ X,Y) – dη(X,φ Y)) ξ + η(Y)h X + η(X)h Y where

hα := 21

ξ φ


Theorem Any almost bi-paracontact manifold admits a unique connection c∇ (called the canonical connection) such that



i) c∇ ξ = 0



i) c∇ ξ = 0

ii) c∇ φ = 32

η⊗h for each =1,2,3



i) c∇ ξ = 0

ii) c∇ φ = 32


iii) cT = dη − 31(dη(φ1∙,φ1∙) + dη(φ2∙,φ2∙) − dη(φ3∙,φ3∙))

− 61( ( ) ( ) ( )111

321NNN −+ ).



i) c∇ ξ = 0

ii) c∇ φ = 32


iii) cT = dη − 31(dη(φ1∙,φ1∙) + dη(φ2∙,φ2∙) − dη(φ3∙,φ3∙))

− 61( ( ) ( ) ( )111

321NNN −+ ).

Furthermore, c∇ is explicitly given by

( )32131

∇∇∇∇ ++=c .


Theorem If the almost bi-paracontact manifold is normal then c∇ satisfies the following properties:



1) c∇ φ1 = c∇ φ2 = c∇ φ3 = 0



1) c∇ φ1 = c∇ φ2 = c∇ φ3 = 0

2) cT = 2dη⊗ξ



1) c∇ φ1 = c∇ φ2 = c∇ φ3 = 0

2) cT = 2dη⊗ξ

3) cR (φ1∙,φ1∙) = cR (φ2∙,φ2∙) = − cR (φ3∙,φ3∙) = − cR



1) c∇ φ1 = c∇ φ2 = c∇ φ3 = 0

2) cT = 2dη⊗ξ

3) cR (φ1∙,φ1∙) = cR (φ2∙,φ2∙) = − cR (φ3∙,φ3∙) = − cR

4) The Ricci tensor is given by

Ricc(X,Y) = −21

trace(Rc(X,Y))

Hence, Ricc is skew-symmetric and

Ricc(φ1∙,φ1∙) = Ricc(φ2∙,φ2∙) = − Ricc(φ3∙,φ3∙) = Ricc


5) c∇∇∇∇ === 321


5) c∇∇∇∇ === 321 6) M is foliated by four mutually transverse Legendre foliations

whose leaves are totally geodesic and admit a flat affine structure


5) c∇∇∇∇ === 321 6) M is foliated by four mutually transverse Legendre foliations

whose leaves are totally geodesic and admit a flat affine structure

7) The 1-dimensional foliation defined by the Reeb vector field ξ is

transversely para-hypercomplex. c∇ is (locally) projectable to the Obata connection on the leaf

space.


The model in the normal case Consider R2n+1 with coordinates (x1,…,xn,y1,…,yn,z) and contact form

∑=

−=n

iiidxydz

1

: .

Examples


∑=

−=n

iiidxydz

1

: .

Set, for each i∈{1,…,n}

ii

yX

∂∂

= , z

yx

Y ii

i ∂∂

+∂∂

= .

Define φ1 Xi := Xi φ1Yi := ‒Yi φ1ξ := 0 φ2 Xi := ‒Yi φ2Yi := ‒Xi φ2ξ := 0 φ3 Xi := Yi φ3Yi := ‒Xi φ3ξ := 0

Then (φ1,φ2,φ3) is a normal almost bi-paracontact structure on R2n+1.

Examples


∑=

−=n

iiidxydz

1

: .

Set, for each i∈{1,…,n}

ii

yX

∂∂

= , z

yx

Y ii

i ∂∂

+∂∂

= .

Define φ1 Xi := Xi φ1Yi := ‒Yi φ1ξ := 0 φ2 Xi := ‒Yi φ2Yi := ‒Xi φ2ξ := 0 φ3 Xi := Yi φ3Yi := ‒Xi φ3ξ := 0

Then (φ1,φ2,φ3) is a normal almost bi-paracontact structure on R2n+1.

The canonical distributions D1+, D1−, D2+, D2− are given by

D1+ = span{X1,…,Xn}, D1‒ = span{Y1,…,Yn}

D2+ = span{X1‒Y1,…,Xn‒Yn}, D2‒ = span{X1+Y1,…,Xn+Yn}

Examples

Further examples Theorem Every contact Riemannian manifold endowed with a Legendre distribution admits an almost bi-paracontact structure.

Examples

Further examples Theorem Every contact Riemannian manifold endowed with a Legendre distribution admits an almost bi-paracontact structure. Therefore we have many examples of almost bi-paracontact manifolds:

- The cotangent sphere bundle of a Riemannian manifold

- Every contact Riemannian manifold such that ξ is an Anosov flow (e.g. the tangent sphere bundle of a negatively curved manifold)

- Any contact metric (κ,µ)-space

Examples

Definition A contact metric ( , )-space is a contact Riemannian manifold (M,φ,ξ,η,g) such that

Rg(X,Y)ξ = κ(η(Y)X – η(X)Y) + µ(η(Y)hX – η(X)hY), ( )

for some κ,µ R, where

h := 21

ξ φ.

( ) is called “(κ,µ)-nullity condition”. If in ( ) κ = 1, then h 0 and the manifold is Sasakian.

Contact metric ( , )-spaces

Definition A contact metric ( , )-space is a contact Riemannian manifold (M,φ,ξ,η,g) such that

Rg(X,Y)ξ = κ(η(Y)X – η(X)Y) + µ(η(Y)hX – η(X)hY), ( )

for some κ,µ R, where

h := 21

ξ φ.

( ) is called “(κ,µ)-nullity condition”. If in ( ) κ = 1, then h 0 and the manifold is Sasakian.

D.E. Blair, T. Koufogiorgos, B.J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 1995


1. The (κ,µ)-nullity condition determines the curvature completely.

Motivations


2. There are non-trivial examples of contact metric (κ,µ)-spaces, the most important being the tangent sphere bundle of a Riemannian manifold of constant sectional curvature c (κ = c(2 – c), µ = –2c).

Motivations



3. Any contact metric (κ,µ)-space is a strongly pseudo-convex CR-manifold.

Motivations




4. In the non-Sasakian case (κ≠1) the contact metric structure is not “projectable”. Thus contact metric ( , )-spaces have no analogues in even dimension.

Motivations




4. In the non-Sasakian case (κ≠1) the contact metric structure is not “projectable”. Thus contact metric ( , )-spaces have no analogues in even dimension.

5. A classification is known

E. Boeckx, A full classification of contact (κ,µ)-spaces, Illinois J. Math. (2000)

Motivations

Theorem (Blair et alt. 1995; C.M. 2010) Let (M,φ,ξ,η,g) be a contact (κ,µ)-manifold. Then κ ≤ 1.

If κ = 1 (M,φ,ξ,η,g) is Sasakian.

If κ < 1 the operators h and φh admit 3 eigenvalues 0, λ, –λ, where λ = κ−1 .


Theorem (Blair et alt. 1995; C.M. 2010) Let (M,φ,ξ,η,g) be a contact (κ,µ)-manifold. Then κ ≤ 1.

If κ = 1 (M,φ,ξ,η,g) is Sasakian.

If κ < 1 the operators h and φh admit 3 eigenvalues 0, λ, –λ, where λ = κ−1 .

The corresponding eigendistributions are Legendre foliations of M and satisfies

T(M) = Dh(λ) ⊕ Dh(-λ) ⊕ Rξ (orthogonal) = Dh(λ) ⊕ Dφh(λ) ⊕ Rξ = Dh(λ) ⊕ Dφh(-λ) ⊕ Rξ = Dh(-λ) ⊕ Dφh(λ) ⊕ Rξ = Dh(-λ) ⊕ Dφh(-λ) ⊕ Rξ = Dφh(λ) ⊕ Dφh(-λ) ⊕ Rξ (orthogonal)


Corollary Every (non-Sasakian) contact metric (κ,µ)-space admits a canonical almost bi-paracontact structure (φ1,φ2,φ3) given by

hφκ

φ−

=11

:1 , hκ

φ−

=11

:2 , φφ =:3 .

Such almost bi-paracontact structure is integrable, non-normal.


Theorem The canonical connection c∇ is a contact connection (i.e. c∇ =

c∇ d = 0), and satisfies

23

312

21

132

132

2

132

21

32

φφ

φφφ

φφ

c

c

c

⊗−=

⊗−+⊗

−=

⊗

−−=

∇

∇

∇


Theorem The canonical connection c∇ is a contact connection (i.e. c∇ =

c∇ d = 0), and satisfies

23

312

21

132

132

2

132

21

32

φφ

φφφ

φφ

c

c

c

⊗−=

⊗−+⊗

−=

⊗

−−=

∇

∇

∇

Conversely …


Theorem Let (φ1,φ2,φ3) be an integrable almost bi-paracontact structure on a contact manifold (M, ) such that c∇ = c∇ d = 0 and

23

312

21

φbφ

φbφaφ

φaφ

c

c

c

⊗=

⊗+⊗=

⊗−=

∇

∇

∇

for some a, b > 0.


Theorem Let (φ1,φ2,φ3) be an integrable almost bi-paracontact structure on a contact manifold (M, ) such that c∇ = c∇ d = 0 and

23

312

21

φbφ

φbφaφ

φaφ

c

c

c

⊗=

⊗+⊗=

⊗−=

∇

∇

∇

for some a, b > 0.

Set g1 := dη(∙,φ1∙) + η⊗η

g2 := dη(∙,φ2∙) + η⊗η

g3 := −dη(∙,φ3∙) + η⊗η and assume that g3 is positive definite.


Then

(φ1,ξ,η,g1), (φ2,ξ,η,g2) are paracontact metric structures

(φ3,ξ,η,g3) is a contact metric structure Moreover, the Levi Civita connections of g1, g2, g3 satisfy a ( , )-nullity condition, where

κ1 = 9/4 a2 − 1, µ1 = 2 − 3b

κ2 = 9/4 (a2 − b2), µ2 = 2

κ3 = 1 − 9/4 b2, µ3 = 2 + 3a


Let us recall that Boeckx introduced the number

−

−=

1

21MI

in order to classify contact metric (κ,µ)-spaces.



−

−=

1

21MI


Theorem Let (M,φ,ξ,η,g) be a contact metric (κ,µ)-space such that |IM|≠1.



−

−=

1

21MI


Theorem Let (M,φ,ξ,η,g) be a contact metric (κ,µ)-space such that |IM|≠1. Then M admits a further integrable almost bi-paracontact structure (φ’1,φ’2,φ’3) given by



−

−=

1

21MI



i) If |IM|>1

( )( )φhφIφ M −+=′

+−−1:

121

121

, hφ−

=′1

1:2 , 213 : φφφ ′′=′



−

−=

1

21MI



i) If |IM|>1

( )( )φhφIφ M −+=′

+−−1:

121

121

, hφ−

=′1

1:2 , 213 : φφφ ′′=′

ii) If |IM|

Let us consider the above almost contact structure

φ'3 := φ’1φ’2.



φ'3 := φ’1φ’2.

Then one proves that

g'3 := −dη(∙,φ’3∙) + η⊗η

is a Riemannian metric, which is compatible with the almost contact structure.



φ'3 := φ’1φ’2.


g'3 := −dη(∙,φ’3∙) + η⊗η

is a Riemannian metric, which is compatible with the almost contact structure. Theorem (C.M. 2010) g'3 is a Sasakian metric!



φ'3 := φ’1φ’2.


g'3 := −dη(∙,φ’3∙) + η⊗η

is a Riemannian metric, which is compatible with the almost contact structure. Theorem (C.M. 2010) g'3 is a Sasakian metric! Corollary Any contact metric (κ,µ)-space such that IM ≠ ±1 admits a compatible Sasakian structure. Consequently, if M is compact, the Betti numbers bi, 1 ≤ i ≤ 2n are even.


1. B. Cappelletti Montano, Bi-Legendrian structures and paracontact

geometry, Int. J. Geom. Meth. Mod. Phys. 2009

2. B. Cappelletti Montano, Bi-paracontact structures and Legendre foliations, Kodai Math. J. 2010.

3. B. Cappelletti Montano, The foliated structure of contact metric (κ,µ)-spaces, Illinois J. Math. (to appear)

4. B. Cappelletti Montano, L. Di Terlizzi, Geometric structures associated to a contact metric (κ,µ)-space, Pacific J. Math. 2010

References

1. B. Cappelletti Montano, Bi-Legendrian structures and paracontact

geometry, Int. J. Geom. Meth. Mod. Phys. 2009

2. B. Cappelletti Montano, Bi-paracontact structures and Legendre foliations, Kodai Math. J. 2010.

3. B. Cappelletti Montano, The foliated structure of contact metric (κ,µ)-spaces, Illinois J. Math. (to appear)

4. B. Cappelletti Montano, L. Di Terlizzi, Geometric structures associated to a contact metric (κ,µ)-space, Pacific J. Math. 2010

OBRIGADO !

References

Bi-paracontact structures on contact manifolds Let (M,φ,ξ,η,g) be a contact metric manifold endowed with a Legendre distribution L. Define Q:=φ(L), Legendre distribution. Set

ψ│L = I, ψ│Q = −I, ψξ = 0.

Then φ1 := φψ, φ2 := ψ, φ3 := φ

is an almost bi-paracontact structure.

Bi-paracontact structures on contact manifolds Every contact manifold (M,η) endowed with a Legendre distribution admits infinitely many bi-Legendrian structures. Infact let L be a Legendre distribution, (φ,ξ,η,g) be an associated contact Riemannian structure and Q=φL. Let ψ be the paracontact structure associated with (L,Q). For any , such that 2 + 2 = 1 we set

ψ , := ψ + φψ.

Then ψ ,

2 = I - η⊗ξ The eingendistributions + ,L and

−,L of ψ , corresponding to 1 and

-1 defines transversal Legendre distributions. Thus ( )−+ ,, ,LL is an almost bi-Legendrian structure on (M,η).

Legendre foliations Example M - smooth manifold, dim(M)=n. T*(M) - the cotangent bundle of M. ω = yidxi - the Liouville form on T*(M), (xi) being coordinates on M, (yi) fiber coordinates. Suppose there exists a function F: T*(M) → [0,+∞[ such that

F(tv) = tF(v), for all t ≥ 0, v∈T*(M).

The unit cotangent bundle with respect to F is the (2n+1)-dimensional hypersurface of T*(M)

SF*(M) = {v∈T*(M) │ F(v) = 1}.

Typical case: (M,g) a Riemannian manifold and take F = ǁ∙ǁ. ( ) is denoted also 1

*( ) and called of ( , ).

Then (S*F(M),η) is a contact manifold, where η = i*(ω) and the foliation FF by fibers of the projection map : SF*(M) → M is a Legendre foliation of (SF*(M),η).

Legendre foliations Theorem (Pang) Any Legendre foliation F on (M,η) is locally equivalent with one of the form FF. Moreover, the equivalence is global provided that the leaves of F are compact and simply connected; in this case F defines a Finsler metric on M.

Finsler metrics A Finsler metric is a function F: T *M → [0,+∞[ such that

F is positively homogeneous: F(tv) = t F(v) for any t > 0 F(v1+v2) ≤ F(v1) + F(v2).

The triangle inequality is equivalent to the convexity of the function F2, and this is again equivalent to the condition that (F2)ij is positive definite on T *M.

Contact Riemannian manifolds Theorem

Let (M,η) be a contact manifold. Then there exist

a Riemannian metric g such that

g(X,ξ) = η(X),

i.e. g(ξ,ξ) = 1 and ξ ⊥ D

a tensor field φ of type (1,1) such that

φ2 = –I + η⊗ξ, g(X,φY) = dη(X,Y).

The geometric structure (φ,ξ,η,g) is called contact Riemannian (or metric) structure.

Anosov flows

Example (Anosov flows) Let (M,φ,ξ,η,g) be a contact Riemannian manifold for which ξ is Anosov, i.e. there exist subbundles Es and Eu (stable and unstable subbundles), invariant along the flow {ωt} of ξ, such that

T(M) = Es ⊕ Eu ⊕ Rξ

and

( )vctavωt −≤∗ exp for all t≥0 and v∈Esx

( )wctawωt exp≤∗ for all t≤0 and w∈Eux,

where a, c > 0 are constants independent of x∈M and v∈Esx, w∈Eux.

Then (Es,Eu) is a flat bi-Legendrian structure on M.

The most notable example of contact manifold for which ξ is Anosov is the tangent sphere bundle of a negatively curved manifold.

Tangent sphere bunlde Let π’: TM → M be the tangent bundle of (M,g). The tangent space to TM at a point (x,u) splits into the direct sum of the vertical subspace VTM(x,u) = ker(π’*|(x,u)) and the horizontal subspace HTM(x,u) with respect to the Levi Civita connection ∇g on M:

T(x,u)TM = VTM(x,u) ⊕ HTM(x,u).

If X∈Γ(TM), Xh and Xv are the horizontal and vertical lift of X on TM. The Sasaki metric gS on TM is defined by

gS(Xh,Yh) = gS(Xv,Yv) = g(X,Y) ◦ π’, gS(Xh,Xv) = 0.

TM admits an almost complex structure J defined by

JXh = Xv, JXv = –Xh.

Tangent sphere bunlde The tangent sphere bundle π: T1M → M is the hypersurface of TM defined by

T1M = {(x,u)∈TM | gx(u,u)=1}.

The geodesic flow of (M,g) is the horizontal vector field of TM defined by

h

i

iu

xuJN

∂

∂=−=′

where (x,u)∈TM and i

i

uuN

∂∂

= . If (x,z)∈T1M then z′ is tangent to

T1M. Let η’ be the 1-form on T1M dual to ′ with respect to gS and φ’ be the (1,1) tensor field given by

φ’X = JX – η’(X)N. Then

(φ’, 2 ′, η’, gS)

is the standard contact metric structure on T1M.

D-homotethic deformations Let (M,φ,ξ,η,g) be a contact metric manifold. By a Da-homothetic deformation we mean the change of structure tensor of the form

a= , a

1= , = , ⊗−+= )1(aaagg

where a > 0. Then

),,,,( gM is also a contact metric manifold.

A ),( -manifold is transformed in a ),( -manifold where

2

2 1a

a −+= ,

a

a 22 −+= .

Let (M,F) be a foliated space and � be a linear connection on M. � is F-projectable if it projects to connections of the local slice space of F. The conditions for this are: a. if σ is a projectable cross section of N(F):=TM/TF, then �Xσ = 0

for all X∈Γ(TF), b. if σ is a projectable cross section of N(F) and X is a basic vector

field, then �Xσ is a projectable cross section of N(F). Here by “projectable cross section” we mean that the vector field that represents σ is basic: σ = [Y]F , Y∈Γ(TM), [Y1]=[Y2] ⇔ Y1–Y2 ∈Γ(TF).

A 3-web is a triple of foliations (F1,F2,F3) on M2n such that for

any , ∈{1,2,3}, ≠ , T(M) = T(F )⊕T(F ).

A para-hypercomplex structure on M2n is given by two anti-commuting product structures I, J and a complex structure K such that IJ = K. Then M2n admits a unique connection, called Obata connection, defined as the unique torsion-free connection preserving the para-hypercomplex structure.

Let N be an n-dimensional manifold and TCN its complexifed tangent bundle. Let H be a complex subbundle of complex dimension l. A CR-manifold of real dimension n and CR-dimension l is a pair (N,H) such that - { }0=∩ pp HH - H is involutive.

Then there exists an unique subbundle D of TN such that DC = HH ⊕ and an unique bundle map J: D → D such that J2 = -I and

H = {X – iJX | X∈D}.

Since φ2 = –I + η⊗ξ and φξ = 0, the eigenvalues of φ are 0, ± i. Thus

DC = D’ ⊕ D’’ where

D’ = {X – iφX | X∈D} D’’ = {X + iφX | X∈D}

Theorem (Tanno) (M2n+1,D’) is (strongly pseudo-convex) CR-manifold if and only if

(∇gXφ)Y = g(X+hX,Y)ξ - η(Y)(X+hX).

BI-PARACONTACT STRUCTURES AND LEGENDRE ...paracontact struct. and φ3,ξ,η) is normal as almost...

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