New PDF release: Geometry IV: Non-regular Riemannian Geometry

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

The ebook encompasses a survey of study on non-regular Riemannian geome­ attempt, conducted regularly via Soviet authors. the start of this course 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 techniques that may be outlined and proof that may be confirmed through measuring the lengths of curves at the floor relate to intrinsic geometry. within the case thought of in differential is outlined by way of specifying its first geometry the intrinsic geometry of a floor primary shape. If the skin F is non-regular, then rather than this type it truly is handy to exploit the metric PF' outlined as follows. For arbitrary issues X, Y E F, PF(X, Y) is the best decrease certain 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 accordingly its intrinsic geometry. in keeping with what we have now stated, the most item of study then looks as a metric house such that any issues of it may be joined via a curve of finite size, and the space among them is the same as the best reduce certain of the lengths of such curves. areas gratifying this are referred to as areas with intrinsic metric. subsequent we introduce metric areas with intrinsic metric pleasurable in a single shape or one other the situation that the curvature is bounded.

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The metric PM on the two-dimensional Riemannian manifold M is the intrinsic metric. For any piecewise smooth path~: [a, b] ..... M of class C 1 in the manifold M, l(~; a, b) is its length with respect to the metric PM' I. Two-Dimensional Manifolds of Bounded Curvature 41 We shall omit the proof of the theorem, as it is rather cumbersome (see Kobayashi and Nomizu (1963), Ch. IV). 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.

Hence it follows that y(t) is a piecewise smooth path of class C. 1f x'(t)-# 0, then since the Jacobian of the function q is non-zero it follows from the equalities indicated that y'(t) -# o. Definition of a two-dimensional Riemannian manifold. Let M be an arbitrary differentiable two-dimensional manifold. Then we shall say that a Riemannian geometry is specified in M, or briefly that M is a Riemannian manifold, if with any piecewise smooth path [a, b] -+ M of class C 1 there is associated a number l(e; a, b) such that the following conditions are satisfied.

We note, however, that those properties of the curvature that we give later can be used, in principle, for its definition. ) With the help of %(X) we can introduce some additive set functions. Namely, let E be an arbitrary Borel set in M. Then if E is bounded (that is, the closure of E is compact), there is defined the integral f f %(p) dS(p) = w(E). E We shall call w(E) the integral curvature of the set E. In addition, we introduce the following quantities: Iwl(E) = f f'%(p), dS(p), E w+(E) = f f %+(p) dS(p), w-(E) = f f %-(p) dS(p).

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