3 Affine Connections with Torsion in (Para-)complexified Structures
41
3.2.2 Nijenhuis Tensor N L and Integrability
On an (even-dimensional) manifold M with an almost (para-)complex structure L,
there is an associated Nijenhuis tensor
N L (X, Y ) = −L
2
[X, Y ] + L[X, LY ] + L[L X, Y ] − [L X, LY ].
(3.7)
When N L = 0, the operator L is said to be integrable. Via the celebrated NewlanderNirenberg theorem, in the complex case L = J , this is the obstruction to turning an
almost complex structure J into a complex structure. In other words, vanishing of
N J is equivalent to M admitting local holomorphic charts, with J as the complex
structure.
The following identity for the Nijenhuis tensor can be easily verified.
Lemma 1 The Nijenhuis tensor N L satisfies
N (L X, Y ) = N (X, LY ) = −L N (X, Y ).
Proof Because L
2
= ±id, it can be taken out of the bracket,
N (L X, Y ) = −L
2
[L X, Y ] + L[L X, LY ] + L[L
2 X, Y ] − [L
2 X, LY ]
= −L
2
[L X, Y ] + L[L X, LY ] + L
3
[X, Y ] − L
2
[X, LY ]
= −L(L[L X, Y ] − [L X, LY ] − L
2
[X, Y ] + L[X, LY ])
= −L N (X, Y ).
Likewise, N (X, LY ) = −L N (X, Y ).
Proposition 2 The Nijenhuis tensor N L can be expressed in the follow forms:
N L (X, Y ) =
T ∇ L (L(L X), Y ) − LT ∇ (L X, Y )
+
T ∇ L (L X, LY ) − LT ∇ (X, LY )
(3.8)
and
N L (X, Y ) =
T ∇ L (L(L X), Y ) + T ∇ L (L X, LY )
− L
T ∇ (L X, Y ) + T ∇ (X, LY )
.
(3.9)
Proof The relationships were derived in Lemma 3 [7]. We prove them below (to
correct certain typos there).
Starting from Eqs. (3.4), (3.5), and (3.6), we have
∇ X (LY ) − ∇ Y (L X) = (∇ X L)Y + L(∇ X Y ) − ((∇ Y L)X + L(∇ Y X ))
= S(X, Y ) + L(∇ X Y − ∇ Y X )
= S(X, Y ) + L([X, Y ] + T
∇
(X, Y ))
= LT
∇
L (X, Y ) + L[X, Y ].
(3.10)