40
J. Zhang and G. Khan
∇
L
X Y = L
−1
(∇ X (LY ))
(3.3)
for any vector fields X and Y on M, i.e., X and Y are elements of (T M). That ∇
L
is indeed an affine connection needs verification, which we omit here. The L-gauge
transformation ∇ → ∇
L of any connection ∇ satisfies
(∇
L 1 )
L 2 = ∇
L 1 ◦L 2 ,
with ◦ denoting the composition of L 1 and L 2 as operators. Identifying ◦ as the group
multiplication operation and the identity transformation L = id as the group identity
element, L-gauge transformation of affine connections form a transformation group
acting on the space of linear connections on M.
It can be seen that ∇ = ∇
L if and only if ∇ L = 0. A connection ∇ is called
(para-)complex if ∇ L = 0. In this paper, we do not generally make the assumption
of (para-)complexity of ∇.
We define a vector-valued skew-symmetric bilinear form S, a (1, 2)-tensor, via
the expression
S(X, Y ) = (∇ X L)Y − (∇ Y L)X,
(3.4)
where
(∇ X L)Y = ∇ X (LY ) − L(∇ X Y ).
The pair (∇, L) is said to be Codazzi-coupled if S = 0.
Recall that the torsion T
∇ of ∇ is defined as:
T
∇
(X, Y ) = ∇ X Y − ∇ Y X − [X, Y ].
(3.5)
We easily derive the identify
S(X, Y ) = L(T
∇
L (X, Y ) − T
∇
(X, Y )).
(3.6)
This leads to the following well-known results.
Proposition 1 (e.g., [10]) Let ∇ be an affine connection, and let L be an arbitrary
tangent bundle isomorphism. Then the following statements are equivalent:
(i) (∇, L) is Codazzi-coupled.
(ii) T
∇
(X, Y ) = T
∇
L (X, Y ).
(iii) (∇
L
, L
−1
) is Codazzi-coupled.
Corollary 1 For the special case of (para-)complex operators L
2
= ±id,
(a) ∇
L
= ∇
L
−1 , i.e., L-conjugate transformation is involutive, (∇
L
)
L
= ∇.
(b) (∇, L) is Codazzi-coupled if and only if (∇
L
, L) is Codazzi-coupled.
Codazzi-coupling of (∇, L) on an almost-(para-)complex manifold (M, L) mirrors Codazzi-coupling of (∇, g) on a Riemannian manifold (M, g).
J. Zhang and G. Khan
∇
L
X Y = L
−1
(∇ X (LY ))
(3.3)
for any vector fields X and Y on M, i.e., X and Y are elements of (T M). That ∇
L
is indeed an affine connection needs verification, which we omit here. The L-gauge
transformation ∇ → ∇
L of any connection ∇ satisfies
(∇
L 1 )
L 2 = ∇
L 1 ◦L 2 ,
with ◦ denoting the composition of L 1 and L 2 as operators. Identifying ◦ as the group
multiplication operation and the identity transformation L = id as the group identity
element, L-gauge transformation of affine connections form a transformation group
acting on the space of linear connections on M.
It can be seen that ∇ = ∇
L if and only if ∇ L = 0. A connection ∇ is called
(para-)complex if ∇ L = 0. In this paper, we do not generally make the assumption
of (para-)complexity of ∇.
We define a vector-valued skew-symmetric bilinear form S, a (1, 2)-tensor, via
the expression
S(X, Y ) = (∇ X L)Y − (∇ Y L)X,
(3.4)
where
(∇ X L)Y = ∇ X (LY ) − L(∇ X Y ).
The pair (∇, L) is said to be Codazzi-coupled if S = 0.
Recall that the torsion T
∇ of ∇ is defined as:
T
∇
(X, Y ) = ∇ X Y − ∇ Y X − [X, Y ].
(3.5)
We easily derive the identify
S(X, Y ) = L(T
∇
L (X, Y ) − T
∇
(X, Y )).
(3.6)
This leads to the following well-known results.
Proposition 1 (e.g., [10]) Let ∇ be an affine connection, and let L be an arbitrary
tangent bundle isomorphism. Then the following statements are equivalent:
(i) (∇, L) is Codazzi-coupled.
(ii) T
∇
(X, Y ) = T
∇
L (X, Y ).
(iii) (∇
L
, L
−1
) is Codazzi-coupled.
Corollary 1 For the special case of (para-)complex operators L
2
= ±id,
(a) ∇
L
= ∇
L
−1 , i.e., L-conjugate transformation is involutive, (∇
L
)
L
= ∇.
(b) (∇, L) is Codazzi-coupled if and only if (∇
L
, L) is Codazzi-coupled.
Codazzi-coupling of (∇, L) on an almost-(para-)complex manifold (M, L) mirrors Codazzi-coupling of (∇, g) on a Riemannian manifold (M, g).