1998/04/30 by V. D. Gladush, R. A. Konoplya · 4 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #Geometric Analysis and Curvature Flows #gr-qc #math.DG
paper · pdf · doi:10.1063/1.532697
published as J.Math.Phys.40:955-979,1999 · 30 pages, RevTeX, aps.sty, some additions and corrections, new references added
arxiv created 1998/11/02 · openalex publication_date 1999/02/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a general approach to the decomposition of the tangent bundle of pseudo-Riemannian manifolds into direct sums of subbundles, and the associated decomposition of geometric objects. An invariant structure Hr defined as a set of r projection operators is used to induce decomposition of the geometric objects into those of the corresponding subbundles. We define the main geometric objects characterizing decomposition. Invariant nonholonomic generalizations of the Gauss–Codazzi–Ricci’s relations have been obtained. All the known types of decomposition (used in the theory of frames of reference, in the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory of stationary spaces, and so on) follow from the present work as special cases when fixing a basis and dimensions of subbundles, and parametrization of a basis of decomposition. Various methods of decomposition have been applied here for the unified multidimensional Kaluza–Klein theory and for relativistic configurations of a perfect fluid. Discussing an invariant form of the equations of motion we have found the invariant equilibrium conditions and their 3+1 decomposed form. The formulation of the conservation law for the curl has been obtained in the invariant form.