University of Rome "La Sapienza", Dipartimento di Fisica, Corso di Fisica Teorico (Ruffini)

Five part notes from lectures given in 3 separate visits.

**Part I:****orthogonal coordinates on flat or constant curvature manifolds with metric, FRW geometry, simplest spacetime splittings, Gaussian normal coords, intrinsic and extrinsic curvature (January 1988)**

(icm1.pdf: 37 pages, 1.163 MEG)

contents:

- What is Cosmology? Overview
- Flat Euclidean/Lorentz geometry for N = 2
- N > 2: (pseudo-)spherical, (pesudo-)cylindrical coordinates
- Orthogonal coordinates on M
^{4}and constant curvature subspaces - Maximally symmetric spaces
- Friedmann-Robertson-Walker spacetimes
- Higher dimensions?
- Special coordinates on de Sitter spacetime
- Special coordinates on anti de Sitter spacetime
- Gaussian normal coordinates and constant curvature, Einstein curvature
- Interpretation of extrinsic curvature
- Arbitrary signature flat spaces and pseudo-orthogonal groups

Part 1 is designed to generalize the student's knowledge of the geometry of special relativity to the case of cosmological spacetimes, from the point of view of metric and coordinates, with some mention of groups of motion. Only geometries of constant curvature are considered.

The geometry of symmetry groups and its application to less symmetric spacetime geometries relevant in cosmological models in 4 or more dimensions will be developed in part 2.

**Part II:****Symmetries and Lie groups (March 1988)**

(icm2.pdf: 29 pages, 731K)

contents:

- Generating vector fields, Killing vector fields, Lie algebras, matrix groups
- Lie group actions (transformation groups), orbits, quotient spaces, homogeneous space
- Lie derivatives, invariance

Part 2 is designed to cover the topic of symmetry groups, in particular, isometry groups (groups of motions of metrics). Metrics with transitive isometry grups (all points equivalent under the actions of the group) are studied as homogeneous spaces useful as spatial sections of a spacetime or as fiber spaces in higher dimensional spacetimes or simply in ordinary gauge theories, both of which are important in current unification schemes. These applications will be covered in part 3.

**Part III: Differential geometry, classical mechanics, matrix groups, rigid body dynamics**

(icm3.pdf: 20 pages, 883K)

contents:

- Another glance at differential geometry: classical mechanics and phase spaces
- Additional useful facts about matrix groups: canonical parametrizations, invariant field computation, adjoint group, SO(3,R) and SU(2)
- Rigid body dynamics

Part 3 studies in more detail the geometry of Lie groups and group invariant metrics and applications to the rigid body problem to bridge the new ideas with an elementary physics style perspective.

**Part IV: Fiber bundles, gauge groups (May 1988)**

(icm4.pdf: 20 pages, 663K)

contents:

- Loose ends from part III: active Euler angles, frame Lagrange derivatives
- What is a fiber bundle? Why bother?
- Kinds of fiber bundles, familiar examples: tensor bundles
- Local trivializations, horizontal and vertical subspaces and connections
- Structural group, sections
- Bundle of frames
- U(1) bundle and electromagnetism
- Nonabelian generalization
- Bundle metric, higher-dimensional Einstein equations, Weyl transformations
- End

Part 4 gives an impression of how fiber bundles work using the tangent and cotangent bundle examples of part 3 and a circle bundle of electromagnetism and the bundle of frames. These are then placed in the context of higher dimensional spacetimes and unified field theories.

This part was written "on the fly" without notes relying on memory so they could easily stand being rewritten someday. Maybe.

(Someday never came...)

**Part V: Anisotropic cosmological models, gravitational dynamics (May 1988)**

(icm5.pdf: 7 pages, 222K)

contents:

- Tying things together
- Anisotropic cosmological models
- Gravitational dynamics, de Witt metric
- Generalization of Kepler and rigid body dynamics
- Bye

Part 5 tries to connect up these ideas with more general ones in general relativity and cosmology.