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Geometry IV - Non-regular Riemannian Geometry (Hardcover, 1993 ed.)
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Geometry IV - Non-regular Riemannian Geometry (Hardcover, 1993 ed.)
Series: Encyclopaedia of Mathematical Sciences, 70
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Total price: R2,957
Discovery Miles: 29 570
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The book contains a survey of research on non-regular Riemannian
geome try, carried out mainly by Soviet authors. The beginning of
this direction oc curred in the works of A. D. Aleksandrov on the
intrinsic geometry of convex surfaces. For an arbitrary surface F,
as is known, all those concepts that can be defined and facts that
can be established by measuring the lengths of curves on the
surface relate to intrinsic geometry. In the case considered in
differential is defined by specifying its first geometry the
intrinsic geometry of a surface fundamental form. If the surface F
is non-regular, then instead of this form it is convenient to use
the metric PF' defined as follows. For arbitrary points X, Y E F,
PF(X, Y) is the greatest lower bound of the lengths of curves on
the surface F joining the points X and Y. Specification of the
metric PF uniquely determines the lengths of curves on the surface,
and hence its intrinsic geometry. According to what we have said,
the main object of research then appears as a metric space such
that any two points of it can be joined by a curve of finite
length, and the distance between them is equal to the greatest
lower bound of the lengths of such curves. Spaces satisfying this
condition are called spaces with intrinsic metric. Next we
introduce metric spaces with intrinsic metric satisfying in one
form or another the condition that the curvature is bounded."
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