Chapter 3
Affine Connections with Torsion
in (Para-)complexified Structures
Jun Zhang and Gabriel Khan
Abstract We investigate integrability conditions for an almost (para-)complex
structure L, L
2
= ±id, on manifolds that admit affine connections carrying torsion in general. The affine connections ∇ under consideration are not assumed to be
(para-)complex ∇ L = 0. Two kinds of torsion matching conditions leading to integrability of L are analyzed, which are generalizations of the first and second canonical
connections along the Gauduchon line. We also discuss (para-)holomorphicity of ∇
encoded by the second condition.
3.1 Introduction
On a differentiable manifold M of even dimension, one can separately consider three
entities all taking inputs from sections (T M) of the tangent bundle T M—an affine
connection ∇ : (T M) × (T M) → (T M), a pseudo-Riemannian metric g :
(T M) × (T M) → CM, and a tangent bundle endomorphism-induced operator
L : (T M) → (T M). Here and below, we use L to denote either an almost
complex operator J, J
2
= −id, or an almost para-complex operator K , K
2
= id
assuming the multiplicity of the ± 1 eigenvalues are equal. We use the term “(para-)
complex” to mean “either complex or para-complex”, corresponding to when L = J
or when L = K .
Enforcing compatibility of L with g
g(L X, Y ) + g(X, LY ) = 0,
(3.1)
J. Zhang (B) · G. Khan
University of Michigan, Ann Arbor, USA
e-mail: junz@umich.edu
G. Khan
e-mail: gkhan@iastate.edu
G. Khan
Iowa State University, Ames, USA
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_3
37
Affine Connections with Torsion
in (Para-)complexified Structures
Jun Zhang and Gabriel Khan
Abstract We investigate integrability conditions for an almost (para-)complex
structure L, L
2
= ±id, on manifolds that admit affine connections carrying torsion in general. The affine connections ∇ under consideration are not assumed to be
(para-)complex ∇ L = 0. Two kinds of torsion matching conditions leading to integrability of L are analyzed, which are generalizations of the first and second canonical
connections along the Gauduchon line. We also discuss (para-)holomorphicity of ∇
encoded by the second condition.
3.1 Introduction
On a differentiable manifold M of even dimension, one can separately consider three
entities all taking inputs from sections (T M) of the tangent bundle T M—an affine
connection ∇ : (T M) × (T M) → (T M), a pseudo-Riemannian metric g :
(T M) × (T M) → CM, and a tangent bundle endomorphism-induced operator
L : (T M) → (T M). Here and below, we use L to denote either an almost
complex operator J, J
2
= −id, or an almost para-complex operator K , K
2
= id
assuming the multiplicity of the ± 1 eigenvalues are equal. We use the term “(para-)
complex” to mean “either complex or para-complex”, corresponding to when L = J
or when L = K .
Enforcing compatibility of L with g
g(L X, Y ) + g(X, LY ) = 0,
(3.1)
J. Zhang (B) · G. Khan
University of Michigan, Ann Arbor, USA
e-mail: junz@umich.edu
G. Khan
e-mail: gkhan@iastate.edu
G. Khan
Iowa State University, Ames, USA
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_3
37