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 booklet incorporates a survey of analysis on non-regular Riemannian geome­ attempt, conducted commonly by means 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 suggestions that may be outlined and evidence that may be confirmed by means 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 skin F is non-regular, then rather than this way 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 reduce 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 therefore its intrinsic geometry. in response to what now we have stated, the most item of analysis then seems as a metric area such that any issues of it may be joined by way of a curve of finite size, and the space among them is the same as the best reduce sure 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 pleasurable in a single shape or one other the that the curvature is bounded.

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Extra info for Geometry IV: Non-regular Riemannian Geometry

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Hence it follows, in particular, that the first derivative x'(t) also has limits to the left and right at these points. For the function x'(t) these limits must be non-zero. [a, b] -+ M a Let M be a differentiable two-dimensional manifold, and parametrized curve in M. We assume that there is an admissible chart qJ: U -+ ~2 such that W) E U for all t E [a, b]. We shall say that is a smooth (piecewise smooth) path of class C if the path x(t) = qJ [e(t)], a ~ t ~ b, on the plane ~2 is smooth (piecewise smooth) of class C.

Let A be the closed set in M consisting of all points of M that correspond to points lying on these curves along which the pasting is carried out. Cutting the space (M, p) along the given set A, we obtain the original collection of two-dimensional manifolds. 4. A Side of a Simple Arc in a Two-Dimensional Manifold. Let M be an arbitrary two-dimensional manifold with boundary, and L a simple arc in M. We introduce here some concepts that enable us to give an exact meaning to words "on a given side of L" or "on the same side of L" (see p.

M, such that each of them is homeomorphic to a half-disc and for the point Xi the boundary of the neighbourhood Vi is divided into two arcs Li and L;. The numbering of the points that constitute p-l(X) and the neighbourhoods Vi can be chosen so that under the pasting by means of which D is obtained from LJ the neighbourhoods Vl and V 2 are pasted to each other along the arcs L~ and L 2 , V 2 and V3 along the arcs L~ and L 3 , and so on, ending with Vm- l and Vm along the arcs L~-l and Lm. Then two cases are possible.

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