Core Module in Differential Geometry
20 credits / 30 lectures
Differential Geometry is one of the oldest mathematical disciplines and originates from the first analytic studies of geometric entities as lines and planes which later turned into what is known now as Classical Geometry of curves and surfaces. Later it developed to Modern Differential Geometry (Differential Geometry of Manifolds). All that evolution happened in close relations to Analysis, Lie Groups and Algebras, Topology, Differential Equations, in particular Dynamical Systems, and some more applied subjects as for example Analytical Mechanics. All these disciplines strongly influenced and still influence each other.
The key concept of the Modern Differential geometry is that of a manifold – a space that is locally as n-dimensional Euclidean space (for finite-dimensional manifolds) or as some Banach space in the infinite dimensional case. The main idea is to study objects on manifolds using local parameterizations (charts) of the manifold, but in such a way that these objects could be given a meaning not depending on those parameterizations. In brief, this leads to the study of various type of tensor fields on manifolds together with the operations with and between them. These types of questions are addressed in the honours course of Differential Geometry and in fact one of the main goals is to develop a generalization of the usual Calculus in the n-dimensional Euclidean space. The issues covered form the foundation on which one could build more specific and more advanced knowledge about manifolds with some particular tensor fields fixed on them (then we say that we endow a manifold with some geometric structure, e.g., Riemann, symplectic, Khaler manifolds etc.).
It should be also underlined the importance of the Differential Geometry for Theoretical Physics. Indeed, the idea of all Physical Theories is to describe the natural phenomena using some parameterizations of some Physical Fields in such a way that general statements about them are not dependent on the parameterizations. Then the Differential Geometry of Manifolds is not only a natural language for the Physical Theories but it is also an indispensable tool without which nowadays it is not possible to study modern Physics -- Field Theories (in particular String Theories) and Gravitation. For the above reasons the honours course of Differential Geometry gives the student the necessary knowledge to study further numerous topics ranging from Pure Mathematics to Theoretical Physics.
- Abstract differentiable manifolds. Charts, parameterizations, differentiable maps on manifolds
- Abstract manifolds versus regular surfaces in Rn . Sub-manifolds. Immersed and imbedded sub-manifolds
- Constructions of differentiable manifolds: a) using action of a group; b) gluing differentiable manifolds.
- Partition of the unity.
- Tangent vectors, the tangent and the cotangent bundles of a differentiable manifold, tangent maps, pull-back maps
- Vector bundles. Tensors, operations with tensors. Tensor bundles and tensor fields.
- The exterior algebra on a vector space. Differential forms and algebraic operations with differential forms.
- The equivalence between the algebra of the vector fields and the algebra of the derivations of the algebra of the smooth functions over a manifold. Lie brackets of vector fields.
- The exterior derivative (Cartan's derivative). The Lie derivative. Basic properties of these operations. Differential calculus on manifolds, derivations of the exterior algebra. The algebra of the derivations of the exterior algebra.
- One-parametric groups of diffeomorphisms and their relations with the Lie bracket and the Lie derivative.
- Manifolds with boundary. Orientation. Stokes theorem.