ON GLOBAL RIEMANN-CARTAN GEOMETRY

12. Pseudo-harmonic vector fields
A vector field ξ on a Riemann-Cartan manifold ( M ,g,∇)
with positive-definite metric tensor g is said to be
pseudo -harmonic if it is a solution of the system of differential equations g ( ∇ ,Y ) − g ( X, ∇ ξ ) 0
0.
[1] Goldberg S.I. On pseudo-harmonic and pseudo-Killing vector in metric manifolds with torsion. The Annals of
Mathematics, 2 nd Ser. 64: 2 (1956), 364-373.
Theorem 12.1. Let a compact oriented n-dimensional (n > 2) Riemann-Cartan manifold ( M ,g,∇)
with positivedefinite
metric tensor g belong to the class Ω 1 ⊕ Ω 2 . If the condition Ric ( ξ , ξ) ≥0
holds for a pseudo-harmonic
⊥
vector field ξ then the hyperdistribution ξ orthogonal to ξ is umbilical.
−−−
ξ ; trace ∇ ξ =
X Y
=
Theorem 12.2. Let a compact oriented n-dimensional (n > 2) Riemann-Cartan manifold ( M ,g,∇)
with positive-
___
definite metric tensor g belongs to the class Ω 2 . If the condition Ric ( ξ, ξ) ≥0
holds for a pseudo-harmonic vector
⊥
field ξ then the hyperdistribution ξ is integrable with maximal totally umbilical manifolds and the metric form of
n
the manifold has the following form in a local coordinate system x 1 ,..., x of a certain chart (U, ψ )
ds
2
= σ
1 n 1 n−1
a b
1 n n n
( x ,...,x ) g ( x ,...,x ) dx ⊗ dx + g ( x ,...,x ) dx ⊗ dx
ab
nn
for a,b
=1,...,n
−1.
27