3 Affine Connections with Torsion in (Para-)complexified Structures
43
Remark. 1st Matching Condition of torsion of ∇ with L, or MC1, resembles the
compatibility condition between a symplectic form ω and L, namely, ω(L X, Y ) =
ω(X, LY ). For this reason, we also refer to MC1 as “torsion-compatibility.” 2nd
Matching Condition of torsion of ∇ with L, or MC2, was referred to as “torsioncoupling” in [7], where a comprehensive study was conducted on how such coupling interacts with Codazzi coupling of (∇, g) in almost (para-)Hermtian manifolds
(M, g, L).
Proposition 3 The Nijenhuis tensor N L vanishes if
(a) MC1 (torsion-compatibility) holds for both ∇ and ∇
L ; or
(b) MC2 (torsion-coupling) holds for either ∇ or ∇
L .
Proof With respect to statement (a), substituting the following torsion-compatibility
conditions
T
∇
(L X, Y ) + T
∇
(X, LY ) = 0
L
2 T
∇
L (X, Y ) + T
∇
L (L X, LY ) = 0
into Eq. (3.9) leads to N L = 0. With respect to statement (b), substituting Eq. (3.13),
the torsion-coupling condition, into Eq. (3.8) leads to N L = 0.
Proposition 4 With respect to the following statements
(i) N L = 0;
(ii) (∇, L) satisfies MC1, i.e., Eq. (3.11);
(iii) (∇
L
, L) satisfies MC1;
(iv) (∇, L) satisfies MC2, i.e., Eq. (3.12) or Eq. (3.13);
(v) (∇
L
, L) satisfies MC2;
we have
(a) (iv) and (v) are equivalent;
(b) (iv) or (v) implies (i);
(c) any two of (i), (ii), (iii) implies the third;
(d) either (iv) or (v) together with (ii) or (iii) imply that
T
∇
= T
∇
L = 0.
Proof That (∇, L) satisfies MC2 means
T
∇
L (L X, Y ) = L
−1 T
∇
(L(L X), Y ) = L
−1
(T
∇
(L
2 X, Y )) = L(T
∇
(X, Y )).
So, that (∇, L) satisfies MC2 is equivalent to that (∇
L
, L) satisfies MC2. Hence (iv)
and (v) are equivalent statements. And each implies N L = 0.
That (ii) plus (iii) imply (i) is what statement (a) of Proposition 3 indicates. To
show (i) and (ii) leads to (iii), we only need to mention that having assumed (ii), the
formula for the Nijenhuis tensor reduces to
43
Remark. 1st Matching Condition of torsion of ∇ with L, or MC1, resembles the
compatibility condition between a symplectic form ω and L, namely, ω(L X, Y ) =
ω(X, LY ). For this reason, we also refer to MC1 as “torsion-compatibility.” 2nd
Matching Condition of torsion of ∇ with L, or MC2, was referred to as “torsioncoupling” in [7], where a comprehensive study was conducted on how such coupling interacts with Codazzi coupling of (∇, g) in almost (para-)Hermtian manifolds
(M, g, L).
Proposition 3 The Nijenhuis tensor N L vanishes if
(a) MC1 (torsion-compatibility) holds for both ∇ and ∇
L ; or
(b) MC2 (torsion-coupling) holds for either ∇ or ∇
L .
Proof With respect to statement (a), substituting the following torsion-compatibility
conditions
T
∇
(L X, Y ) + T
∇
(X, LY ) = 0
L
2 T
∇
L (X, Y ) + T
∇
L (L X, LY ) = 0
into Eq. (3.9) leads to N L = 0. With respect to statement (b), substituting Eq. (3.13),
the torsion-coupling condition, into Eq. (3.8) leads to N L = 0.
Proposition 4 With respect to the following statements
(i) N L = 0;
(ii) (∇, L) satisfies MC1, i.e., Eq. (3.11);
(iii) (∇
L
, L) satisfies MC1;
(iv) (∇, L) satisfies MC2, i.e., Eq. (3.12) or Eq. (3.13);
(v) (∇
L
, L) satisfies MC2;
we have
(a) (iv) and (v) are equivalent;
(b) (iv) or (v) implies (i);
(c) any two of (i), (ii), (iii) implies the third;
(d) either (iv) or (v) together with (ii) or (iii) imply that
T
∇
= T
∇
L = 0.
Proof That (∇, L) satisfies MC2 means
T
∇
L (L X, Y ) = L
−1 T
∇
(L(L X), Y ) = L
−1
(T
∇
(L
2 X, Y )) = L(T
∇
(X, Y )).
So, that (∇, L) satisfies MC2 is equivalent to that (∇
L
, L) satisfies MC2. Hence (iv)
and (v) are equivalent statements. And each implies N L = 0.
That (ii) plus (iii) imply (i) is what statement (a) of Proposition 3 indicates. To
show (i) and (ii) leads to (iii), we only need to mention that having assumed (ii), the
formula for the Nijenhuis tensor reduces to