| Abstrakt: |
We study the properties of real realizations of holomorphic linear connections over associative commutative algebras $$ \mathbb{A} $$ m with unity. The following statements are proved. If a holomorphic linear connection ∇ on M n over $$ \mathbb{A} $$ m ( m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ℝ of the Lie algebra of all affine vector fields of the space ( M , ∇ℝ) is no greater than ( mn)2 − 2 mn + 5, where m = dimℝ $$ \mathbb{A} $$ , $$ n = dim_\mathbb{A} $$ M n , and ∇ℝ is the real realization of the connection ∇. Let ∇ℝ =1 ∇ ×2 ∇ be the real realization of a holomorphic linear connection ∇ over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor 1 R ≠ 0, 2 R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space ( M , ∇ℝ) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( a M n , a ∇) ( a = 1, 2). [ABSTRACT FROM AUTHOR] |
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