Mathematics and its Applications (Dordrecht). 393. Dordrecht: Kluwer Academic Publishers. xviii, 393 p. Dfl. 295.00; $ 195.00; £ 120.00 (1997).

The author presents some aspects related to the geometries subordinated to the simple Lie groups from the classes $A_n$, $B_n$, $C_n$, $D_n$ as well as to the exceptional Lie groups $G_2$, $F_4$, $E_6$, $E_7$, $E_8$. The book is based on some earlier books by the author such as [Multidimensional Spaces (Russian), Nauka, Moskow (1966;

Zbl 0143.43801), Non-Euclidean Spaces (Russian), Nauka, Moskow (1969;

Zbl 0176.18901)], some unpublished manuscripts, the author’s research results and his teaching experience at various universities in the USSR and USA. It can be considered as a reference book for the problems arising in various non-Euclidean geometries.
After an introductory chapter on algebraic and topological structures, differentiable manifolds, Riemannian geometry, topological and Lie groups and homogeneous and symmetric spaces, the author presents, in Chapter I, some aspects from the theory of algebras and Lie groups. There are discussed the properties of commutative associative algebras, noncommutative associative algebras, alternative algebras, Lie algebras and Lie groups, Jordan and elastic algebras and of the linear representations of simple Lie groups. In Chapter II some results from affine and projective geometries are presented, such as: affine and projective spaces over algebras and their real and complex interpretations, affine and projective transformations, lines, $m$-planes and hyperplanes in affine and projective spaces, hyperquadrics, linear complexes, projective configurations, finite geometries. Next, in Chapters III, IV and V, the author presents various aspects of the Euclidean, pseudo-Euclidean, conformal, pseudoconformal, elliptic, hyperbolic, pseudoelliptic, pseudohyperbolic, quasielliptic, quasihyperbolic and quasi-Euclidean geometries, such as: the groups of motions and similitudes, $m$-planes, polyhedra, hyperquadrics, hyperspheres, sliding vectors, trigonometry, space forms, applications to physics, etc. Similar problems are considered for the symplectic and quasisymplectic geometries as well as for the geometries subordinated to the exceptional Lie groups $G_2$, $F_4$, $E_6$, $E_7$ and $E_8$. Then the properties from the metasymplectic geometries are obtained studying the noncompact groups of the classes $F_4$, $E_6$, $E_7$, $E_8$.
As said at the beginning, the book is useful as a reference work for various notions and results in the studied geometries.