GeometryThe second book in the series deals with the calculus of vector and tensor-valued differential forms, though founded in the cardinal ideas of Felix Klein and É. Cartan on the foundations of geoometry. In other words, this is differential geometry. A few chapters have already been written. Here is the philosophy that inspires it. A student of pre-Riemannian clasical differential geometry (mainly the theory of curves and surfaces) will be well served by the study of at least some parts of D. Struick’s beautiful book “Classical differential geometry”. The next step should be the first four chapters of another beauty, “Elements of the Tensor Calculus” by A. Lichnerowicz. There one gets the right idea about what affine and Euclidean spaces are, and learns the tensor calculus the right way. Also (in the fourth chapter), one studies Euclidean space in almost the right way. One should not go beyond the fourth chapter, Riemannian geometry, but rather into the study of affine and Euclidean space from the perspective of the affine and Euclidean group. One differentiates the equations for the general elements of these groups, acting on frames (therefore, the moving frame). One then looks at the differentiated equations from the oppossite perspective: integrability. That is the Cartan way. In doing so, one learns in flat spaces what torsion and affine curvature are. Affine and Euclidean spaces have affine curvature and torsion; it simply happens that they are zero. Metric curvature does not exist in affine spaces in general; it exists in Euclidean spaces, where it happens to be zero. The understanding of the theory of affinely and Euclideanly connected spaces is then an easy walk. The study of the “flat cases” is esential for an understanding of the evolution of geometry, starting, say, with Riemann in the mid 19th century: Riemann versus Klein, unification by Cartan, etc.In order to get the most of that smooth transition from flat to generalized spaces, one has to think of tangent vectors and tensors in a passive role, i.e. as equivalence classes of curves rather than as differential operators, even if these are isomorphic structures. For Cartan, the concept of vector field is not the modern one, which he calls an infinitesimal transformation and which Cartan calls a Lie operator. Also, Cartan works with bundles, seldom with sections of the bundle. Combine the study of flat spaces from a perspective of integrability with the bundle approach and what do you get: you learn to live in the bundles, where eveything is far more transparent. That is the Cartan world of differential geometry. |
AuthorsJose Vargas and Doug Torr will make every effort to respond to questions on their papers and the subject of Cartan-Kähler Unification. |