The Einstein Challenge for Differential Geometry - Short Outlook

Short outlook Weyl and Cartan: geometry and physics – For both mathematicians discussed here, Weyl and Cartan, the study of general differential geometry was closely intertwined with physical considerations. – The challenge of general relativity was crucial for their mathematical and conceptual innovations. Also the backreaction of their respective works on physical theories was considerable (in particular in the long run). – But there were considerable differences in I I the way they considered the new PoS, and the relationship between math. considerations and physics. PoS – Weyl and Cartan – Weyl: PoS the conceptual challenge to consequently “infinitesimalize” differential geometric structures. Guiding question: Which automorphism groups for SR and GR? Inbuilt principle: Unique (torsion free) affine connection. Clue of his contribution gauge idea. – Cartan: PoS the challenge to close the gap between Kleinian type homogeneous spaces and Riemannian geometry. Guiding question: How to implement infinitesimal translational elements in the connections? That was done for connections in a variety of groups. Clue of his contribution translational curvature, torsion. I Key ideas gauge geometry/fields and torsion not at all “postmature”, but – if anything of that kind – rather “premature”: . PoS – Weyl and Cartan – Weyl: PoS the conceptual challenge to consequently “infinitesimalize” differential geometric structures. Guiding question: Which automorphism groups for SR and GR? Inbuilt principle: Unique (torsion free) affine connection. Clue of his contribution gauge idea. – Cartan: PoS the challenge to close the gap between Kleinian type homogeneous spaces and Riemannian geometry. Guiding question: How to implement infinitesimal translational elements in the connections? That was done for connections in a variety of groups. Clue of his contribution translational curvature, torsion. I Key ideas gauge geometry/fields and torsion not at all “postmature”, but – if anything of that kind – rather “premature”: . PoS – Weyl and Cartan – Weyl: PoS the conceptual challenge to consequently “infinitesimalize” differential geometric structures. Guiding question: Which automorphism groups for SR and GR? Inbuilt principle: Unique (torsion free) affine connection. Clue of his contribution gauge idea. – Cartan: PoS the challenge to close the gap between Kleinian type homogeneous spaces and Riemannian geometry. Guiding question: How to implement infinitesimal translational elements in the connections? That was done for connections in a variety of groups. Clue of his contribution translational curvature, torsion. I Key ideas gauge geometry/fields and torsion not at all “postmature”, but – if anything of that kind – rather “premature”: . Mathematics and physics – For Weyl the mathematical generalizations in GR and differential geometry were motivated, sometime heuristically guided, by philosophical and/or physical considerations. – Cartan’s work in differential geometry developed “organically” from the symbolical tools of the French tradition in geometry (Darboux etc.) and his own contributions to differential forms and Lie groups. Physics was a challenge/trigger for going ahead and a rich field of application, rather than a motivational force driving his mathematical innovations. – Not to forget: There were many mathematicians worldwide contributing to the interplay of differential geometry and GR. – The field is open for further studies! –