ON GLOBAL RIEMANN-CARTAN GEOMETRY

For a compact Riemann-Cartan manifold ( M ,g,∇)
of the class Ω 1 ⊕ Ω 2 , we have
Then the following theorem is true.
s
⎛
⎝
( ) ( )
( 1) b
M = s M −∫
⎜ S + ( n − ) ( 2
2 2
)
M
2
S
b
2
⎞
⎟dV
⎠
. (9.3)
Theorem 9.2. The complete scalar curvatures s ( M ) and s ( M ) of Riemannian compact oriented manifold ( M ,g ) and
a compact oriented Riemann-Cartan manifold ( M,g ,∇)
of class Ω 1 ⊕ Ω 2 are related by the inequality ( M ) s( M )
s ≤ .
For dimM
≥3
, the equality is possible if the connection ∇ coincides with the Levi-Civita connection ∇ of the metric
g, for n = 2, if ∇is a semi-symmetric connection.
For a compact Weitzenböck manifold ( M,g ,∇)
of the class Ω 1 ⊕ Ω 2 the inequality ( M ) ≥ 0
formulate (see Lemma 7.1)
s holds. Therefore we can
Corollary 1. There are not Weitzenböck connections ∇ of the class Ω 1 ⊕ Ω 2 on a compact Riemannian manifold
with ( M ) < 0
s .
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