Akivis, M. A.; Goldberg, V. V. Semiintegrable almost Grassmann structures. (English) Zbl 0921.53006 Differ. Geom. Appl. 10, No. 3, 257-294 (1999). The authors study locally semiflat (also called semiintegrable) almost Grassmann structures. They establish necessary and sufficient conditions for an almost Grassmann structure to be \(\alpha\)- or \(\beta\)-semiintegrable. These conditions are expressed in terms of the fundamental tensors of almost Grassmann structures. Since the authors were not able to prove the existence of locally semiflat almost Grassmann structures in the general case, they give many examples of \(\alpha\)- and \(\beta\)-semiintegrable structures, mostly four-dimensional. For all examples they find systems of differential equations of the families of integral submanifolds \(V_\alpha\) and \(V_\beta\) of the distributions \(\Delta_\alpha\) and \(\Delta_\beta\) of plane elements associated with an almost Grassmann structure. For some examples they were able to integrate these systems and find closed form equations of submanifolds \(V_\alpha\) and \(V_\beta\). Reviewer: V.V.Goldberg (Newark/New Jersey) Cited in 1 ReviewCited in 4 Documents MSC: 53A40 Other special differential geometries 53A60 Differential geometry of webs 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:almost Grassmann structure; locally semiintegrable; locally semiflat; webs PDF BibTeX XML Full Text: DOI References:  Akivis, M.A.; Goldberg, V.V., Conformal differential geometry and its generalizations, (1996), Wiley-Interscience New York · Zbl 0863.53002  Akivis, M.A.; Goldberg, V.V., On the theory of almost Grassmann structures, (), 1-37 · Zbl 0940.53021  Akivis, M.A.; Shelekhov, A.M., Geometry and algebra of multidimensional three-webs, (1992), Kluwer Dordrecht · Zbl 0792.53009  Cartan, É.; Cartan, É., LES espaces á connexion conforme, (), 2, 747-797, (1923), also · JFM 50.0493.01  Dhooghe, P.F., Grassmannianlike manifolds, (), 147-160, Leuven/Brussels, 1992 · Zbl 0848.53005  Dhooghe, P.F., Grassmannian structures on manifolds, Bull. belg. math. soc. Simon stevin, 1, 1, 597-621, (1994) · Zbl 0934.53025  Goldberg, V.V., 4-tissus isoclines exceptionnels de codimension deux et de 2-rang maximal, C.R. acad. sci. Paris Sér. I math., 301, 11, 593-596, (1985) · Zbl 0579.53015  Goldberg, V.V., Isoclinic webs W (4, 2, 2) of maximum 2-rank, (), 168-183, Peniscola 1985  Goldberg, V.V., Theory of multicodimensional (n + 1)-webs, (1988), Kluwer Dordrecht · Zbl 0668.53001  LeBrun, C., anti-self-dual Riemannian 4-manifolds, (), 81-94 · Zbl 0830.53023  Taubes, C.H., The existence of anti-self-dual conformal structures, J. diff. geom., 36, 163-253, (1992) · Zbl 0822.53006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.