ON GLOBAL RIEMANN-CARTAN GEOMETRY
ON GLOBAL RIEMANN-CARTAN GEOMETRY ON GLOBAL RIEMANN-CARTAN GEOMETRY
We consider a Riemann-Cartan manifold ( M ,g,∇) of the class Ω 1 which is characterized by the conditions ( 2 ) ( 3 = ) S = 0 S that is equal to S b 3 ∈C ∞ Λ M . For this condition the identity (9.1) can be rewritten as s = s − ( 1) S 2 . Hence we have s ≤ s , and equality is possible only if ∇ = ∇. The following theorem holds. Theorem 9.1. The scalar curvatures s and s of the metric connection ∇ and of the Levi-Civita connection ∇ of an n-dimensional Riemannian-Cartan manifold ( M,g ,∇) of the class Ω 1 satisfy the inequality s ≤ s . The equality s = s is possible only if ∇ = ∇. Let ( M ,g,∇) be a compact Riemann-Cartan manifold, we define its complete scalar curvature as the number s ( M ) = ∫ s d V as an analogue of the complete scalar curvature s( M ) = ∫ s d V M of a Riemannian manifold. M The dependence between the complete scalar curvatures s(M) and s (M) is described in the following formula ∫ ( ) ( 1) b 2 ( ) ( ) ( ) ( 2 = − + − ) b 2 − ( 3 ) b 2 s M s M S 2 ň 2 S 2 S dV . (9.2) In particular, for the Weitzenböck connection ∇ we have the integral identity M ∫ b 2 ( S ) ( 1) b 2 ( ) ( ) ( 2 = + − ) b 2 − ( 3 M S 2 ň 2 S 2 ) s dV . M 20
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 . 21
- Page 1 and 2: ON GLOBAL RIEMANN-CARTAN GEOMETRY P
- Page 3 and 4: T. Kibble and D. Sciama have found
- Page 5 and 6: Classification of known kinds of me
- Page 7 and 8: The development of geometry of metr
- Page 9 and 10: We know that S b ∈C ∞ Λ 2 M
- Page 11 and 12: 4. The class Ω 1 ⊕ Ω 2 of Cappo
- Page 13 and 14: 6. Examples of Riemann-Cartan manif
- Page 15 and 16: The classification of almost Hermit
- Page 17 and 18: 8. Green theorem for a Riemann-Cart
- Page 19: 9. Scalar and complete scalar curva
- Page 23 and 24: The differential equation ( ∇ ξ,
- Page 25 and 26: 11. Vanishing theorems for pseudo-K
- Page 27 and 28: 12. Pseudo-harmonic vector fields A
We consider a Riemann-Cartan manifold ( M ,g,∇)<br />
of the class Ω 1 which is characterized by the conditions<br />
( 2 ) ( 3<br />
=<br />
)<br />
S = 0<br />
S that is equal to S<br />
b 3<br />
∈C<br />
∞ Λ M . For this condition the identity (9.1) can be rewritten as s = s −<br />
( 1) S<br />
2<br />
. Hence<br />
we have<br />
s ≤ s , and equality is possible only if ∇ = ∇. The following theorem holds.<br />
Theorem 9.1. The scalar curvatures s and s of the metric connection ∇ and of the Levi-Civita connection ∇ of an<br />
n-dimensional Riemannian-Cartan manifold ( M,g ,∇)<br />
of the class Ω 1 satisfy the inequality s ≤ s . The equality s = s is<br />
possible only if<br />
∇ = ∇.<br />
Let ( M ,g,∇)<br />
be a compact Riemann-Cartan manifold, we define its complete scalar curvature as the number<br />
s ( M ) =<br />
∫<br />
s d V as an analogue of the complete scalar curvature s( M ) = ∫ s d V<br />
M<br />
of a Riemannian manifold.<br />
M<br />
The dependence between the complete scalar curvatures s(M) and s (M) is described in the following formula<br />
∫<br />
( )<br />
( 1) b<br />
2<br />
( ) ( ) ( ) ( 2<br />
= − + −<br />
) b<br />
2<br />
−<br />
( 3 ) b<br />
2<br />
s M s M S 2 ň 2 S 2 S dV . (9.2)<br />
In particular, for the Weitzenböck connection ∇ we have the integral identity<br />
M<br />
∫<br />
b<br />
2<br />
( S )<br />
( 1) b<br />
2<br />
( ) ( ) ( 2<br />
= + −<br />
) b<br />
2<br />
−<br />
( 3<br />
M S 2 ň 2 S 2<br />
)<br />
s dV .<br />
M<br />
20
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