Supervisor: Maido Rahula, emeriitprofessor, TÜ matemaatika instituut
Vladimir Balan, professor, Bukaresti Polütehniline Ülikool, Rumeenia
Kaarin Riives-Kaagjärv, füüs.-mat.kand., Eesti Maaülikool, Eesti
One of the most significant tools in differential geometry and global analysis, in continuous environment mechanics and dynamical systems is the notion of vector field. In the current thesis the vector field is considered as follows. First, the vector field can be interpreted as a differential operator and as a stop-frame of a movement. Second, a manifold can be projected by vector field onto the space of invariants, and this involves classification of singularities. Third, the vector field itself can be projectable, which allows to obtain the necessary symmetries and invariants from other structures (for instance, from the jet space). The central notion of this thesis is the Lie derivatives of tensor fields. The Lie differentiation technique is developed in nonholonomic (noncoordinate) basis, where an important role is played by derivation formulas together with the nonholonomy object. There is introduced an integration of tensor fields as a reverse process to the Lie differentiation. This definition generalizes the notion of integral of an ordinary function. A few geometrical examples included in the text clarify the topic being discussed. The Lie derivatives are infinitesimal versions of representation of a diffeomorphism group on tensor fields. In this regard, the structure of a tangent group of a Lie group is studied, and is shown how general linear group GL(n,R) acts on itself by left and right translations, and by interior automorphisms (adjoint representation). In this thesis the structure of the space of infinite jets is studied. In particular, analogously to the process, when a smooth map induces an infinite jet, the composition of smooth maps induces a composition of jets. The problem is how to define the jet composition in such a way that the definition does not depend on a choice of maps. The problem leads to another question: how the corresponding total differentiation operators and Cartan forms are related under the jet composition. The answers are given in the form of convenient recurrence formulas.