Journal of the Ramanujan Mathematical Society

Ravi S. Kulkarni ¹

Affiliations
1 Harish-Chandra Researh Institute, Allahabad, US

Abstract

The transformations in classical geometries often come equipped with natural spatial and numerical invariants. Moreover, although they are infinitely many transformations, their “dynamical types” are finite in number. While not attempting to define the notion of a dynamical type, we attempt to explain these phenomena by relating them to the centralizer-conjugacy classes, or z-classes, of elements in the auto-morphism groups of these geometries. We prove a fibration theorem 2.1 for an orbit class in a general group-action, and specialize it to the z-classes. It describes two set-theoretic fibrations of each orbit class, in particular a z-class. The base and the fiber in these fibrations of a z-class provide a partial explanation of the spatial and numerical invariants. The possible finiteness of dynamical types is related to the possible finiteness of z-classes.We bring out the implicit role played by the field of real numbers in the assertion of finiteness of dynamical types of transformations in classical geometries. The analysis shows that the finiteness of dynamical types is expected to hold in the analogues of classical geomeries defined over other local fields. Two examples, namely i) the plane Euclidean geometry, and ii) the semisimple operators on a vector space over an arbitrary field, are worked out in detail to illustrate the general ideas.

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Ravi S. Kulkarni ¹

Affiliations
1 Harish-Chandra Research Institute, Allahabad, IN

Abstract

We define curvatures of a non-singular parallelepiped in a real vector space equipped with a non-degenerate quadratic form of type (p, q), and relate them to curvature functions of parametrized curves in a pseudo-Riemannian manifold. These functions are patterned over the classical Frenet-Serret curvature functions as extended by Blaschke, [1]. We consider arithmetic properties of these and other curvature functions. For example, we prove that if M is a non-singular real algebraic variety defined over a subfield 𝔽 of ℝ, equipped with a pseudo-Riemannian metric also defined over 𝔽, then the Riemannian sectional curvature at a point and the section defined over 𝔽 lies in 𝔽.

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