Notion of almost analytic 1-forms on almost

product manifolds

Dorian Bogdan Stoica and Ioana -Adriana UrdeaScientific Coordinator :Lect. univ. dr. IDA Cristian

May 24, 2014

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1 Introduction

Fix a triple (M,F, ω) with Mm a smooth m-dimensional manifold , F atensor field of (1, 1)-type on M and ω a differentiable 1-form i.e ω ∈ Ω(M)

2 1-forms almost analytic on a almost complexmanifolds

Definition 2.1

• Tensor field F defines an almost complex structure on M if :

F 2 = −I, (1)

where I is the identity.

• The F -conjugate of ω is the 1-form :

ω = ωF := ω F−1 = −ω F (2)

It followsω = −ω F = −ω (3)

To the pair (F, ω) we associate a 2-form defined by :

ΩF,ω(X,Y ) := dω(FX, Y ) + dω(X,Y ) (4)

Definition 2.2 :The 1-form ω is called almost F -analytic relativ to the almostcomplex structure F if ΩF,ω = 0

We consider Ω1(M,F ) the manifold of all almost F analytic forms. In thefollowing we use the convention :

dω(X,Y ) = X(ω(Y ))− Y (ω(X))− ω([X,Y ]) (5)

Proposition 2.1 The 1-form ω is almost F -analytic if and only if its F -conjugate ω is almost F -analytic. If ω is almost F - analytic then ω is closed ifand only if ω is closed .

Proof : using (1) , (3) and (4) we obtain :ΩF,ω(X,Y ) := dω(FX, Y )− dω(X,Y ) = ΩF,ω(FX, Y )ΩF,ω(FX, Y ) := −dω(X,Y )− dω(FX, Y ) = −ΩF,ω(X,Y )

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For ω almost F -analytic ⇐⇒ ΩF,ω(X,Y ) = 0 ⇐⇒ dω(FX, Y ) +dω(X,Y ) = 0 , multiplying the last relation with F we obtain : dω(X,Y ) −dω(FX, Y ) = 0 ⇐⇒ ΩF,ω(X,Y ) = 0

Now , we presume ω almost F -analytic ⇐⇒ ΩF,ω(X,Y ) = 0 ⇐⇒dω(X,Y ) − dω(FX, Y ) = 0 .Multiplying the last relation with F we obtain:dω(FX, Y ) + dω(X,Y ) = 0 ⇐⇒ ΩF,ω(X,Y ) = 0

On a manifold endowed with an almost complex structure , the form of theNijenhuis tensor field of F is :

NF (X,Y ) := [FX,FY ]− F [X,Y ]− F [X,FY ]− [X,Y ] (6)

We obtain the following identities :

NF (FX, Y ) = −FNF (X,Y ) = NF (x, FY )

NF (FX,FY ) = −NF (X,Y ) (7)

Proposition 2.2 If ω is almost F -analytic then :

ω NF = ω NF (8)

Proof :Let ω almost F -analytic. Using (5) , ω F = −ω , ω F = ω , from

ΩF,ω(X,Y ) = dω(FX, Y ) + ω(X,Y ) = 0 ⇐⇒

FX(ω(Y ))− Y (ω(FX))− ω(ω([FX, Y ]) +X(ω(Y ))− ω([X,Y ]) = 0 ⇐⇒

ω(F [X,Y ]) = −X(ω(Y ))− FX(ω(Y )) + ω([FX, Y ]) (9)

Putting X → FX , Y → FY in (9) by direct calculus we obtain :

(ω NF )(X,Y ) = ω([FX,FY ])− ω(F [FX.Y ])− ω(F [X,FY ])− ω[X,Y ]

= ω([FX,FY ])+FX(ω(Y ))−X(ω(Y ))+ω([X,Y ])+X(ω(FY ))+FX(ω(FY ))−ω([FX,FY ])− ω[X,Y ] = 0

By Proposition 2.1 ω is also almost F -analytic and the relation(ω NF )(X,Y ) = 0 follows in a similar manner starting from :

ΩF,ω(X,Y ) = dω(FX, Y )− dω(X,Y ) = 0.

The operators OF , O∗F : Ω2(M)→ Ω2(M) defined by :

OF (ρ)(X,Y ) :=1

2[ρ(X,Y )− ρ(FX,FY )],

O∗F (ρ)(X,Y ) :=

1

2[ρ(X,Y ) + ρ(FX,FY )]

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are the Obata operators associated to F . Using these operators we obtain aclassification of 2-forms with respect to F :

Definition2.3 The 2-form ρ is aalled F -pure if O∗F (ρ) = 00 respectively

F -hybrid if OF (ρ) = 0

Proposition 2.3 If F is an almost complex structure and ω is almostF -analytic form then the 2-forms dω , dω are F -pure.

Proof :Setting X → FX in (9) we have :

ω(F [FX, Y ]) = −FX(ω(Y )) +X(ω(Y ))− ω([X,Y ]) ⇐⇒ (10)

X(ω(Y )) + FX(ω(FY )) = ω([X,Y ]) + ω(F [FX, Y ])

Changing X with Y in the relation above we have :

Y (ω(X)) + FY (ω(FX)) = −ω([X,Y ])− ω(F [X,FY ]) (11)

Decreasing (11) from (10) we obtain :

2dω+ω([X,Y ])+2dω(FX,FY )+ω([FX,FY ]) = 2ω([X,Y [)+ωF ([FX, Y ]+[X,FY ])

which means 4O∗F (dω) = −ω NF = 0. By analogy we obtain : 4O∗

F (dω) =−ω NF = 0.

3 1-forms almost analytic on a almost productmanifolds

Definition 3.1

• Tensor field F defines an almost complex structure on M if :

F 2 = I, (12)

where I is the identity.

• The F -conjugate of ω is the 1-form :

ω = ωF := ω F−1 = ω F (13)

It followsω = −ω F = ω (14)

To the pair (F, ω) we associate a 2-form defined by :

ΩF,ω(X,Y ) := dω(FX, Y )− dω(X,Y ) (15)

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Definition 3.2:The 1-form ω is called almost F -analytic relativ to the al-most product structure F if ΩF,ω = 0.

We consider Ω1(M,F ) the manifold of all almost F analytic forms. In thefollowing we use the convention :

dω(X,Y ) = X(ω(Y ))− Y (ω(X))− ω([X,Y ]) (16)

Proposition 3.1 The 1-form ω is almost F -analytic if and only if its F -conjugate ω is almost F -analytic. If ω is almost F - analytic then ω is closed ifand only if ω is closed .

In the context of the almost product structure , the expression the Nijenhuistensor fiels of F becomes :

NF (X,Y ) := [FX,Fy]− F [FX, Y ]− F [X,FY ] + [X,Y ] (17)

and , as in the case of almost complex structure we obtain the following :

NF (FX, Y ) = −FNF (X,Y ) = NF (X,FY )

NF (FX,FY ) = −NF (X,Y ) (18)

Proposition 3.2 If the 1-form ω is almost F -analytic , then :

ω NF = ω NF = 0 (19)

The Obata operators associated to the almost product structure F aredefined in the same way as in the case of the almost complex structure OF , O

∗F :

Ω2(M)→ Ω2(M) ,

OF (ρ)(X,Y ) :=1

2[ρ(X,Y )− ρ(FX,FY )],

O∗F (ρ)(X,Y ) :=

1

2[ρ(X,Y ) + ρ(FX,FY )].

Proposition 3.3 If F is an almost product structure and ω is almostF -analytic form then the 2-forms dω , dω are F -hybrid.

Proof :Let ω almost analytic. We have

ΩF,ω(X,Y ) = 0 ⇐⇒ dω(FX, Y )− dω(X,Y ) = 0 ⇐⇒

FX(ω(Y ))−Y (ω(FX))−ω([FX, Y ])−X(ω(Y ))−Y (ω(X))−ω([X,Y ]) = 0 ⇐⇒

ω(F [X,Y ]) = C(ω(Y ))− FX(ω(Y )) + ω([FX, Y ]).

Putting X → FX in the last relation , we have :

ω(F [FX, Y ]) = FX(ω(Y ))−X(ω(Y )) + ω([X,Y ]) ⇐⇒

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X(ω(Y ))− FX(ω(FY )) = ω([X,Y ])− ω(F [FX, Y ]). (20)

Changing X with Y in (20) we obtain :

Y (ω(X))− FY (ω(FX)) = −ω([X,Y ]) + ω(F [X,FY ]). (21)

Descreasing (21) from (20) :

4OF (dω) = ω NF = 0

4OF (dω) = ω NF = 0

⇐⇒ dω and dω are F -hybrid .

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