##
**The theory of sprays and Finsler spaces with applications in physics and biology.**
*(English)*
Zbl 0821.53001

Fundamental Theories of Physics. 58. Dordrecht: Kluwer Academic Publishers,. xvi, 308 p. (1993).

This book has been written by three well-known specialists, two mathematicians: P. L. Antonelli (Canada), M. Matsumoto (Japan) and a physicist: R. S. Ingarden (Poland). In a uniform manner they present in equal measures the geometrical theory of Finsler spaces based on the concept of spray and significant applications to thermodynamics, optics, ecology, evolution and developmental biology.

Although the geometry of Finsler spaces was investigated by many reputed geometers, it had the status of a half-daughter of differential geometry over 70 years. A large volume of calculation has shaded the profound geometrical meaning of this theory. The Finsler spaces appeared to be significant in the last two decades after their geometry had been based on the notion of fibre bundle. Their theoretical importance was to be foreseen from their beginnings, since these spaces are the most natural generalization of Riemannian spaces. This fact has led to the nowadays penetrating applications of Finsler spaces to mechanics, theoretical physics, ecology, biology, etc.

Consequently, the initiative of the authors to write a book providing a good introduction into Finsler geometry and significant applications of it in physical theories and biology, has to be very positively appreciated. In brief, the contents of the book, divided into six chapters, references and index, is as follows.

Some preliminary facts regarding fibre bundles (tangent, cotangent, jets), the notion of spray and the connections determined by a spray are included in Chapter 0. The notions of Finsler metric, Finsler space and remarkable classes of Finsler spaces (Randers, \((\alpha,\beta)\)-metric, 1-form metric, etc.) are treated in Chapter 1. A large theory of the connections derived from the fundamental function \(F(x,y)\) of a Finsler space can be found in Chapter 2. Here the torsion, the curvature, the parallelism, the canonical spray derived from \(F\), the famous connections due to Cartan, Berwald and other linear connections associated to \(F\) are clearly, rigorously and minutely treated. New classes of Finsler spaces: Berwald spaces, locally Minkowski spaces, spaces of constant curvature, which are indispensable for applications, are discussed in Chapter 3. Here special attention is paid to the 2-dimensional Finsler spaces which are useful in Finslerian biology. Chapter 4 contains important applications to physics. The main idea is to establish physical meanings for \(F(x,y)\). Thus it is shown that in the theory of crystals \(F(x,y)\) correspond to the phase of an optical wave. In physiological optics, \(F(x,y)\) is the usual length. In mechanics, it is the energy and in thermodynamics, it is entropy. So, the main applications are found in geometrical optics of anisotropical media, physiological optics and binocular visual space, electronic optics with a magnetic field, dissipative mechanics, and thermodynamics. These applications also suggest new research in field theory and particle physics. The last chapter is devoted to applications in biology. It is provided a nonlinear generalization of Huxley and Needham’s concept of allometry for colonial animals or plants. The Finslerian theory of Volterra-Hamilton systems is applied to the interactions in sessile communities. The chapter begins with chemically mediated predator and herbivore interactions. A mathematical theory is developed in order to describe the evolution of fossil invertebrates like polymorphic bryozoans of certain genera of corals. The concept of production predictability as measured by Jacobi stability is used. The positive scalar curvature of a Berwald space is involved in the stability of a community. The ecological theory and allometric form of the Wilson ergonomic theory via Finsler spaces of Wagner type are treated.

In conclusion, the authors have achieved an excellent book, well-balanced between geometric theory and its applications to theoretical physics and mathematical biology. It is attractive and interesting for a large circle of scientists and graduate students.

Although the geometry of Finsler spaces was investigated by many reputed geometers, it had the status of a half-daughter of differential geometry over 70 years. A large volume of calculation has shaded the profound geometrical meaning of this theory. The Finsler spaces appeared to be significant in the last two decades after their geometry had been based on the notion of fibre bundle. Their theoretical importance was to be foreseen from their beginnings, since these spaces are the most natural generalization of Riemannian spaces. This fact has led to the nowadays penetrating applications of Finsler spaces to mechanics, theoretical physics, ecology, biology, etc.

Consequently, the initiative of the authors to write a book providing a good introduction into Finsler geometry and significant applications of it in physical theories and biology, has to be very positively appreciated. In brief, the contents of the book, divided into six chapters, references and index, is as follows.

Some preliminary facts regarding fibre bundles (tangent, cotangent, jets), the notion of spray and the connections determined by a spray are included in Chapter 0. The notions of Finsler metric, Finsler space and remarkable classes of Finsler spaces (Randers, \((\alpha,\beta)\)-metric, 1-form metric, etc.) are treated in Chapter 1. A large theory of the connections derived from the fundamental function \(F(x,y)\) of a Finsler space can be found in Chapter 2. Here the torsion, the curvature, the parallelism, the canonical spray derived from \(F\), the famous connections due to Cartan, Berwald and other linear connections associated to \(F\) are clearly, rigorously and minutely treated. New classes of Finsler spaces: Berwald spaces, locally Minkowski spaces, spaces of constant curvature, which are indispensable for applications, are discussed in Chapter 3. Here special attention is paid to the 2-dimensional Finsler spaces which are useful in Finslerian biology. Chapter 4 contains important applications to physics. The main idea is to establish physical meanings for \(F(x,y)\). Thus it is shown that in the theory of crystals \(F(x,y)\) correspond to the phase of an optical wave. In physiological optics, \(F(x,y)\) is the usual length. In mechanics, it is the energy and in thermodynamics, it is entropy. So, the main applications are found in geometrical optics of anisotropical media, physiological optics and binocular visual space, electronic optics with a magnetic field, dissipative mechanics, and thermodynamics. These applications also suggest new research in field theory and particle physics. The last chapter is devoted to applications in biology. It is provided a nonlinear generalization of Huxley and Needham’s concept of allometry for colonial animals or plants. The Finslerian theory of Volterra-Hamilton systems is applied to the interactions in sessile communities. The chapter begins with chemically mediated predator and herbivore interactions. A mathematical theory is developed in order to describe the evolution of fossil invertebrates like polymorphic bryozoans of certain genera of corals. The concept of production predictability as measured by Jacobi stability is used. The positive scalar curvature of a Berwald space is involved in the stability of a community. The ecological theory and allometric form of the Wilson ergonomic theory via Finsler spaces of Wagner type are treated.

In conclusion, the authors have achieved an excellent book, well-balanced between geometric theory and its applications to theoretical physics and mathematical biology. It is attractive and interesting for a large circle of scientists and graduate students.

Reviewer: R.Miron (Iaşi)

### MSC:

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

53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |

53Z05 | Applications of differential geometry to physics |

92D15 | Problems related to evolution |

78A05 | Geometric optics |

82B99 | Equilibrium statistical mechanics |

92D25 | Population dynamics (general) |

92D40 | Ecology |

92D50 | Animal behavior |