Download Geometry IV: Non-regular Riemannian Geometry by Yu. G. Reshetnyak (auth.), Yu. G. Reshetnyak (eds.) PDF

By Yu. G. Reshetnyak (auth.), Yu. G. Reshetnyak (eds.)

The ebook features a survey of study on non-regular Riemannian geome­ test, performed often by way of Soviet authors. the start of this path oc­ curred within the works of A. D. Aleksandrov at the intrinsic geometry of convex surfaces. For an arbitrary floor F, as is understood, all these innovations that may be outlined and proof that may be verified by way of measuring the lengths of curves at the floor relate to intrinsic geometry. within the case thought of in differential is outlined through specifying its first geometry the intrinsic geometry of a floor basic shape. If the outside F is non-regular, then rather than this kind it truly is handy to take advantage of the metric PF' outlined as follows. For arbitrary issues X, Y E F, PF(X, Y) is the best reduce sure of the lengths of curves at the floor F becoming a member of the issues X and Y. Specification of the metric PF uniquely determines the lengths of curves at the floor, and for that reason its intrinsic geometry. in keeping with what we've acknowledged, the most item of study then seems as a metric house such that any issues of it may be joined via a curve of finite size, and the gap among them is the same as the best reduce certain of the lengths of such curves. areas pleasing this situation are referred to as areas with intrinsic metric. subsequent we introduce metric areas with intrinsic metric gratifying in a single shape or one other the that the curvature is bounded.

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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.

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