By William Kingdon Clifford, Hanna Nencka

This quantity is the outgrowth of a convention dedicated to William ok. Clifford entitled, 'New tendencies in Geometrical and Topological Methods', which was once held on the collage of Madeira in July and August 1995. the purpose of the convention was once to compile lively staff in fields associated with Clifford's paintings and to foster the alternate of rules among mathematicians and theoretical physicists. Divided into 6 one-day classes, every one consultation was once dedicated to a particular point of Clifford's paintings. This quantity is an try and convey the Clifford legacy in a brand new viewpoint to a bigger neighborhood of mathematicians and physicists. New options, rules, and effects stemming from Clifford's paintings are mentioned. every one article within the booklet is a self-contained paper that used to be awarded at or submitted to the convention

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Let Ij be the arc [Xj' X j + 1 ] of this splitting, where j = 1, 2, ... , m. We specify arbitrarily a topological map cp of the domain G onto the disc B(O, 1) in ~2. We orient r so that the following condition is satisfied. If p(t), a ~ t ~ b, is a right parametrization of r, then the function x(t) = cp[p(t)], a ~ t ~ b, defines in ~2 the positive orientation of the boundary of the disc B(O, 1). G. Reshetnyak 46 this, each of the arcs Ij is oriented. Along each of the arcs Ij we specify an orientation of M by means of the following agreement.

If (j): U -+ 1R2 is a chart in a two-dimensional manifold M, then V = (j)(U) is an open set in 1R2. U is called the domain of definition of the chart (j), and V is called its range. Suppose that p E U and (j)(p) = (t 1, t 2). The numbers t 1, t2 are called the coordinates of the point p E U with respect to the given chart. The charts (j)1: U1 -+ 1R2, (j)2: U2 -+ 1R2 of a two-dimensional manifold Mare said to be overlapping if U1 n U2 is not empty. In this case there are defined open sets in 1R2: G1 = (j)1(U 1 n U2) and G2 = (j)2(U1 n U2), and topological maps (J = (j)2 0 (j)1 1: G1 -+ G2,.

The rule for transforming the coefficients of the metric tensor on going over to another coordinate system. In the classical handbooks on Riemannian geometry the rule for transforming the coefficients of the metric tensor is included in the definition of a Riemannian manifold. We show how to derive this rule, starting from the definition given here. Let cp: U -+ 1R2 and 1jJ: U -+ 1R2 be two admissible coordinate systems with a common domain of definition U. Suppose that the differential quadratic forms 2 2 2 L L giix1,X2) dX i dx i=l j=l j, 2 L L hdYl' Y2) dYk dYI' k=l 1=1 are representations of the line element of the Riemannian manifold by means of the charts cp and IjJ respectively.