Topics in Differential Geometry

  title={Topics in Differential Geometry},
  author={Peter W. Michor},
Manifolds and vector fields Lie groups and group actions Differential forms and de Rham cohomology Bundles and connections Riemann manifolds Isometric group actions or Riemann $G$-manifolds Symplectic and Poisson geometry List of symbols Bibliography Index. 
Riemannian foliation with exotic tori as leaves
We construct smooth fiber bundles such that the fibers are exotic tori and the total space has finite abelian fundamental group. This gives examples of a Riemannian foliation on a closed manifold
Equivariant de Rham cohomology: theory and applications
This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on
The Flow of Gauge Transformations on Riemannian Surface with Boundary
We consider the gauge transformations of a metric G-bundle over a compact Riemannian surface with boundary. By employing the heat flow method, the local existence and the long time existence of
Optimal Symplectic Connections on Holomorphic Submersions
The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of
Hamiltonian structures for projectable dynamics on symplectic fiber bundles
The Hamiltonization problem for projectable vector fields on general symplectic fiber bundles is studied. Necessary and sufficient conditions for the existence of Hamiltonian structures in the class
Reductions of Dynamics on Second Iterated Bundles of Lie Groups
We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian
Hörmander's condition for normal bundles on spaces of immersions
Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and
An 1-differentiable cohomology induced by a vector field
A new cohomology, induced by a vector field, is defined on pairs of differential forms (1– differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an
As the third of our series of papers on differential geometry of microlinear Frolicher spaces is this paper devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that
Variational calculus on Lie algebroids
It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of


Curvature and Characteristic Classes
Differential forms and cohomology.- Multiplicativity. The simplicial de rham complex.- Connections in principal bundles.- The chern-weil homomorphism.- Topological bundles and classifying spaces.-
Momentum Maps and Hamiltonian Reduction
Introduction.- Manifolds and smooth structures.- Lie group actions.- Pseudogroups and groupoids.- The standard momentum map.- Generalizations of the momentum map.- Regular symplectic reduction
Differential Geometry, Lie Groups, and Symmetric Spaces
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric
Lie Groupoids and Lie Algebroids in Differential Geometry
Introduction 1. The algebra of groupoids 2. Topological groupoids 3. Lie groupoids and Lie algebroids 4. The cohomology of Lie algebroids 5. An obstruction to the integrability of transitive Lie
Lie groups, Lie algebras, and their representations
1 Differentiable and Analytic Manifolds.- 2 Lie Groups and Lie Algebras.- 3 Structure Theory.- 4 Complex Semisimple Lie Algebras And Lie Groups: Structure and Representation.
Lie group valued moment maps
We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of
Gauge Theory for Diffeomorphism Groups
We consider fibre bundles without structure group and develop the theory of connections, curvature, parallel transport, (nonlinear) frame bundle, the gauge group and it’s action on the space of
Basic differential forms for actions of Lie groups. II
A section of a Riemannian G-manifold M is a closed submanifold Σ which meets each orbit orthogonally. It is shown that the algebra of G-invariant differential forms on M which are horizontal in the
Natural operations in differential geometry
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.
Lie Groups, Lie Algebras, Cohomology and some Applications in Physics
Preface 1. Lie groups, fibre bundles and Cartan calculus 2. Connections and characteristic classes 3. A first look at cohomology of groups and related topics 4. An introduction to abstract group