Natural operations in differential geometry

  title={Natural operations in differential geometry},
  author={Ivan Kol{\'a}ř and Jan Slov{\'a}k and Peter W. Michor},
I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII. Further Applications.- VIII. Product Preserving Functors.- IX. Bundle Functors on Manifolds.- X. Prolongation of Vector Fields and Connections.- XI. General Theory of Lie Derivatives.- XII. Gauge Natural Bundles and Operators.- References.- List of symbols.- Author index. 
Natural differential operations on manifolds: an algebraic approach
Natural algebraic differential operations on geometric quantities on smooth manifolds are considered. A method for the investigation and classification of such operations is described, the method of
Geometric analysis on real analytic manifolds
The continuity, in a suitable topology, of algebraic and geometric operations on real analytic manifolds and vector bundles is proved. This is carried out using recently arrived at seminorms for the
Natural lifting of connections to vertical bundles
Abstract. First we study the flow prolongation of projectable vector fields with respect to a bundle functor of order (r,s,q) on the category of fibered manifolds. Usign this approach, we constructs
Second order connections on some functional bundles
Abstract. We study the second order connections in the sense of C. Ehresmann. On a fibered manifold Y, such a connection is a section from Y into the second non-holonomic jet prolongation of Y. Our
In classical field theory, the composite fibred manifolds Y → Σ → X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This
Second order natural Lagrangians on coframe bundles
We study the structure of second order natural Lagrangians on the bundle of linear coframes F X over an n-dimensional manifold X . They are identified with the corresponding differential invariants
Differential bundles and fibrations for tangent categories
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth


Differential Geometry of Foliations: The Fundamental Integrability Problem
I. Differential Geometric Structures and Integrability.- 1. Pseudogroups and Groupoids.- 2. Foliations.- 3. The Integrability Problem.- 4. Vector Fields and Pfaffian Systems.- 5. Leaves and
Cohomology of Infinite-Dimensional Lie Algebras
1. General Theory.- 1. Lie algebras.- 2. Modules.- 3. Cohomology and homology.- 4. Principal algebraic interpretations of cohomology.- 5. Main computational methods.- 6. Lie superalgebras.- 2.
Bundle functors on Fibred manifolds
We give a proof of the regularity of bundle functors on a certain class of categories over manifolds and a description of all bundle functors on fibred manifolds with fixed dimensions of bases and
Prolongation of vector fields to jet bundles
The main result of the present paper is that all natural operators transforming every projectable vector eld on a bred manifold Y into a vector eld on its r-th jet prolongation J r Y are the constant
Integral curves of derivations
We integrate, by a constructive method, derivations of even degree on the sections of an exterior bundle by families of Z2-graded algebra automorphisms, dependent on a real parameter, and which
Graded derivations of the algebra of differential forms associated with a connection
The central part of calculus on manifolds is usually the calculus of differential forms and the best known operators are exterior derivative, Lie derivatives, pullback and insertion operators.
The Geometry of Jet Bundles
Introduction 1. Bundles 2. Linear bundles 3. Linear operations on general bundles 4. First-order jet bundles 5. Second-order jet bundles 6. Higher-order jet bundles 7. Infinite jet bundles