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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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