By Maks A. Akivis, Alexander M. Shelekhov (auth.)
'The publication is warmly instructed to everybody doing examine within the similar fields.'Deutschen Mathematiker-Vereinigung 97:21 1993
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Extra info for Geometry and Algebra of Multidimensional Three-Webs
The structure equations of a two-dimensional web were derived by Blaschke [Bl 55]. The term "the web curvature" was also introduced in [Bl 55]. 5 Parallelizable and group three-webs were considered by Chern [C 36b] as webs on which the closure conditions T and R hold (in [C 36b] these conditions are called conditions (D) and (P) respectively). 11 can be also found in [C 36b]. However, in [C 36b] it is given in more complicated and less geometric form than in this book. 6. 58) for calculation of the torsion and curvature tensors of a web were found in [AS 71a], although expressions of these tensors close to that in these formulas can be also found in [C 36b].
A i +a~ai )(wk Awl+w k Awl+wk Awl). 31), we can write ... = bikl + 2aiikajmll)' bikl . 91 ) 1. 91) are the structure equations of the middle connection * r. W 3. Later we will use the another type of connection, generated by the web Set = (X, A",) in the manifold X. 94) Denote by 1'12 the connection defined in the manifold X by the forms p~. 95) respectively. 27) have the form: and for the connection 1'31 they are of the form: dw i + dw i = = 1 2 dw i 3 . • • • Ie w' 1\ WJ•. 95), we see that r wi ...
91) are the structure equations of the middle connection * r. W 3. Later we will use the another type of connection, generated by the web Set = (X, A",) in the manifold X. 94) Denote by 1'12 the connection defined in the manifold X by the forms p~. 95) respectively. 27) have the form: and for the connection 1'31 they are of the form: dw i + dw i = = 1 2 dw i 3 . • • • Ie w' 1\ WJ•. 95), we see that r wi ... 96) w} + a~k(Pfk + q't k ). 97) J where u, v = 1,2, ... , 2r, and 0; = We denote this pencil by /(W).