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**General theory of irregular curves. Transl. from the Russian by L. Ya. Yuzina.**
*(English)*
Zbl 0691.53002

There is a comprehensive introduction to the direct geometric methods in the theory of curves, developed by the authors and their coworkers since 1946. It is based on the classical investigations due to Jordan, Frechet, Lebesgue, Rado et al., and extends their research to other geometric quantities. The curves under consideration are assumed to be continuous only, and the geometric quantities are defined by polygonal approximations in principle. The presentation of the subjects is self- contained and can be understood with minor mathematical prerequisites only. Naturally the first two chapters are devoted to the classical topics, a discussion of the possibilities to define a curve and the problem how to measure the length of a curve. For the general theory, continuous curves are considered, having their image in spaces which are as general as possible. For curves in metric spaces the natural distance and convergence are established. Then the length of a curve is introduced and the classical criteria for its rectifiability are derived in the cases where the ambient spaces are Euclidean or Lipschitz manifolds.

Chapter III discusses the notion of a tangent in this general context. One-sidedly smooth curves are defined, and their rectifiability is shown. Several criteria for the existence of tangents are exhibited. A description in terms of contingencies is given too. The rest of this chapter is devoted to the discussion of the generalization of the tangent image of a smooth curve. Then, as a preparation for the subsequent chapters, a short introduction to the fundamentals of integral geometry is presented: The manifold of k-dimensional subspaces of a real vector space of dimension n, realizations of this space by embeddings into Euclidean spaces, invariant measures and integrals on this space, descriptions of the length of a curve in terms of integral geometry, etc.

The turn or total curvature (translated as ‘integral curvature’) of a curve is defined in chapter V as the (possibly infinite) supremum of the angle sums of inscribed polygons. After some justification of this notion the main consequences of the finiteness of this entity are demonstrated: Such curves are one-sidedly smooth, hence rectifiable, vanishing curvature characterizes straight lines, the usual estimate for the total curvature of a closed curve is generalized to the case under consideration, the turn is given as the length of the tangent indicatrix, etc. The finiteness of the turn is described in terms of integral geometry, and some theorems are derived using this terminology. This is followed by length estimates, convergence theorems and other interesting results on the turn of a curve. The next chapter generalizes this theory to curves in spheres.

The goal of the chapters VII and VIII is the extension of the Frenet theory for smooth curves in 3-space to the irregular situation. To this purpose (one-sided) osculating planes are defined, and criteria for their existence are developed, clearly going beyong the finiteness of the turn. Then the total torsion is introduced and several of its properties are presented, including a discussion of the planar case which is not trivial in this general situation. Finally the last chapter recollects the preceding considerations and presents the Frenet theory promised at the beginning.

This monograph contains a lot of interesting results, combining elementary geometry, differential geometry and integral geometry with methods from real analysis. It can be used as a good reference for results on irregular curves, though its presentation sometimes is a little difficult to understand. This partially may be a consequence of the bad English translation, where rather frequently definite and indefinite articles are mixed up. This may lead to difficulties for the unexperienced reader.

Chapter III discusses the notion of a tangent in this general context. One-sidedly smooth curves are defined, and their rectifiability is shown. Several criteria for the existence of tangents are exhibited. A description in terms of contingencies is given too. The rest of this chapter is devoted to the discussion of the generalization of the tangent image of a smooth curve. Then, as a preparation for the subsequent chapters, a short introduction to the fundamentals of integral geometry is presented: The manifold of k-dimensional subspaces of a real vector space of dimension n, realizations of this space by embeddings into Euclidean spaces, invariant measures and integrals on this space, descriptions of the length of a curve in terms of integral geometry, etc.

The turn or total curvature (translated as ‘integral curvature’) of a curve is defined in chapter V as the (possibly infinite) supremum of the angle sums of inscribed polygons. After some justification of this notion the main consequences of the finiteness of this entity are demonstrated: Such curves are one-sidedly smooth, hence rectifiable, vanishing curvature characterizes straight lines, the usual estimate for the total curvature of a closed curve is generalized to the case under consideration, the turn is given as the length of the tangent indicatrix, etc. The finiteness of the turn is described in terms of integral geometry, and some theorems are derived using this terminology. This is followed by length estimates, convergence theorems and other interesting results on the turn of a curve. The next chapter generalizes this theory to curves in spheres.

The goal of the chapters VII and VIII is the extension of the Frenet theory for smooth curves in 3-space to the irregular situation. To this purpose (one-sided) osculating planes are defined, and criteria for their existence are developed, clearly going beyong the finiteness of the turn. Then the total torsion is introduced and several of its properties are presented, including a discussion of the planar case which is not trivial in this general situation. Finally the last chapter recollects the preceding considerations and presents the Frenet theory promised at the beginning.

This monograph contains a lot of interesting results, combining elementary geometry, differential geometry and integral geometry with methods from real analysis. It can be used as a good reference for results on irregular curves, though its presentation sometimes is a little difficult to understand. This partially may be a consequence of the bad English translation, where rather frequently definite and indefinite articles are mixed up. This may lead to difficulties for the unexperienced reader.

Reviewer: Bernd Wegner

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53A04 | Curves in Euclidean and related spaces |

53C65 | Integral geometry |

53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |

26A99 | Functions of one variable |