This short bibliography consists of few books to help a physicist become immersed in Cartan’s geometry and Kähler’s calculus. Cartan’s method of moving frames is essential for an effective approach to both undertakings. It in turn requires as prerequisites some familiarity with geometry using vector fields geometry of curves and surfaces and (b) being acquainted with what is usually known as the exterior calculus of (scalar-valued) differential forms. These prerequisites can be obtained either in parallel or in any order.
A very nice and classic book in the basic geometry of curves and surfaces is Classical Differential Geometry by D. Struick. One should read it selectively, which means: read from each chapter stopping before getting bored or bogged down. It is not worth spending a great effort on old techniques.
Differential forms can be learned from two different perspectives: the dominant and modern one, where differential forms are defined as antisymmetric multilinear functions of vectors, and Kähler’s, where differential forms are functions of hypersurfaces, i.e. integrands to be evaluated on those hypersurfaces. Anybody who knows the vector calculus almost knows about the last ones, but has not been made aware of it. However, the exterior calculus is the same in both perspectives; non-mathematicians will find easier the approach of the second perspective.
For the first perspective, our “recommendation” is the dedicated book Differential Forms by Henri Cartan (son of Élie). For the second perspective, one of the authors of this webpage has written a down-to-earth version. It constitutes the second chapter of a book in the process of being written. The chapter can be downloaded from this website [Link to the chapter presented in Pane 4.2]. After reading it, mathematically minded readers could always resort to Chapter X of the book Principles of Mathematical Analysis by Walter Rudin, which also is written from the second perspective, but much more rigorously. The typical visitor to this web site may not be interested in that level of rigor.
The next step would consist in using differential forms in some basic geometric problems. There is not a book that follows in sufficient detail what we think is the most logical approach, namely the characterization with differential forms of what affine and Euclidean spaces are, as well as their frame bundles. This is not only for the purpose of understanding what the structure of those spaces is, but also because the study of surfaces is the study of Euclidean 3-space itself to which one has added the restrictive condition that defines the surface, and which adds difficulty to the study of affine space. Differential forms allow one to define those spaces from a differential point of view and to see their defining equations as integrability conditions (equations of structure of the affine or Euclidean space) of a certain system of differential equations (known as the connection equations), .
For brief treatments of that step of the learning program, see the third (its last) chapter of H. Cartan’s aforementioned book, where the emphasis is, however, on the theory of curves and surfaces with the moving frame method in Euclidean 3-space. It does not require that the reader had studied the previous two chapters, if one already knows how to work with differential forms. Chapter IV of H. Flanders book Differential Forms with Applications to the Physical Sciences is marginally more comprehensive in those regards. Both of those books are very economically priced.
For a highly readable presentation of the whole gamut of concepts of vector spaces, Euclidean vector spaces, affine spaces, Euclidean vector spaces, tensor algebra, duality, linear forms, multilinear forms, etc the best reference we know of is the first four chapters of the book Elements of the Tensor Calculus by A. Lichnerowicz. The reader should perhaps stop somewhere in the middle of Chapter IV to avoid contamination with those formulas which start to take one in the wrong path. We mean that, for instance, it is wrong to consider the torsion as a 3-tensor and the affine curvature as a 4-tensor. They are respectively a vector-valued differential 2-form and a (1, 1)-tensor-valued differential 2-form. Readers who have read the recommended chapters of the aforementioned books by H. Cartan, Flanders and Lichnerowicz should then be able to follow the digressions in our papers on affine space Euclidean space and their non-holonomic generalizations, digressions required for an understanding of the Cartan – Einstein unification program. They are contained in sections 2 and 3 of our paper 22 [LINK] and sections 1-3 of paper 24 [LINK]. Readers may then try to follow our pre-Finslerian papers 1-3.
Cartan’s geometries are based on basic Lie group and Lie algebra theory. A comprehensive book on these subjects is R. Gilmore’s Lie Groups, Lie Algebras and Some of Their Applications (Warning: their “Ú” symbol should not be interpreted as Clifford product; the author does not claim so, but does not warn either). The Kähler calculus is based on Clifford algebra. The most lucid simple book on this algebra is G. Casanova’s L’Algèbre Vectorielle.
Long Bibliography: some References Used in this Work Cartan (1904, 1905; 21, 22; II) Sur la structure des groupes infinis de transformations
Cartan (1908; 26; II). Les sous-groupes des groupes continus de transformations
Cartan (1910a; 29; III.1) Sur les développables isotropes et la méthode du tri
Cartan (1910b; 31; III.1) La structure des groupes de transformations continus et la théori du tri
Cartan, (1915; 46; III.2). La théorie des groupes continus et la géométrie (Encyclop. Sc. Math.). This article seems to be an adaptation of some German paper or book by G. Fano from Turin, but publication details of Fano’s work are not provided.
Cartan (1922; 56; III.1) Paper on Einstein’s equations.
Cartan (1922a, b, c, d, e; 57-61; III.1). Notes
Cartan, (1923, 24, 25; 66, 69, 80; III.1). Papers on the theory of affine connections
Cartan (1923, 1924, 19250. La theorie des Connexions Affines. . Oeuvres Completes.
Cartan, E. (1924), Les recentes generalizations de la notion d’ espace. Oeuvres Completes.
Cartan, É. (1924). Bulletin des Sciences. Mathématiques. 48, 294. Reprinted: Oeuvres Complètes (1984), Éditions du CNRS, Paris, Volume III, Part 1.
Cartan (1924a; 71; III.1) Les récentes généralizations de la notion d’espace.
Cartan (1925) La théorie des groupes et les recherches récentes de géométrie différentielle. L’Enseignement mathematique, 24, 1-18.
Cartan, (1927; 105; I). La théorie des groupes et la géométrie (L”Enseignement math.)
Cartan, É. and A. Einstein, Letters on Absolute Parallelism, 1929-1932, R. Debever, ed. (Princeton University Press, Princeton, New Jersey, 1979).
Cartan (1930; 124; III.2 ) Notice historique …teleparallelism.
Cartan (1931; 130; III.2 ) Le parallelism.absolu et la th uni
Cartan, É. (1932). Revue de Métaphysique et de Morale,. Reprinted: Oeuvres Complètes, Volume III, Part 1.
Cartan, É. (1934 ?)“Les Systèmes Différentieles Extérieurs et leurs Applications Géométriques (Herman, Paris, 1971).
Cartan, E. (1934). Les Espaces de Finsler. Reprinted in “Exposes de geometrie”, Hermann, Paris, 1971.
Cartan, É. (1934). Actualités scientifiques et industrielles, 72. Reprinted in Exposés de Géometrie, Hermann, Paris, 1971.
Cartan (1935; 144; III.2) La Methode du repere mobile
Cartan, É. (1936). Comptes Rendus Congrès International de Oslo I, 92. Reprinted: Oeuvres Complètes, Volume III, Part 2.
Cartan (1936; 151, III.2 ) congreso de Oslo.
Cartan, É. (1937). Séminaire de Mathematiques, exposé de Jan.11. Reprinted: Oeuvres Complètes, Volume II.
Cartan, (1937). La théorie des groupes finis et continus et la géométrie différentielle …
Chern, S. S. (1948). Science Reports Tsing Hua Univ. 4, 85.
Chern, S. S. (1992). Comptes Rendus Acad. Sci. Paris, 314, 757.
Chern, S. S. (1996). Contemporary Mathematics 196, 51.
Clifton, Y. H. (1966) Journal of Mathematics and Mechanics 16(6), 569.
Debever, R., Editor (1979) Elie Cartan- Albert Einstein, Letters on Absolute Parallelism 1929-1932, Princeton University Press, Princeton.
Einstein, A. (1930) “Auf die Riemann-Metrik und den Fern-Parallelismus Gegründete Einheitliche Feldtheorie, Math. Annalen, Vol. 102, pp. 685-697.
Einstein, A. (1930) “Théorie Unitaire du Champ Physique”, Ann. Inst. Henri Poincaré, Vol. 1, No.1, pp. 1-24.
Feynman, R. P., Leighton, R.B, Sands, M., The Feynman Lectures on Physics (Addison-Wesley, Reading, Massachusetts, 1963).
Finsler, P. (1918). Über Curven und Flächen in Allgemeinem Räumen, Dissertation, Göttingen. Reprinted (1951): Birkhäuser, Basel.
Gardner, R (1989)
Kähler, E. (1960) “Innerer and Äusserer Differentialkalkül”, Abh.Dtsch. Akad. Wiss. Berlin, Kl. Math., Phy. Tech 4, 1;
Kähler, E. (1961) “Die Dirac-Gleichung”, Abh.Dtsch. Akad. Wiss. Berlin, Kl. Math., Phy. Tech 1, 1; (1962). Rendiconti di Matematica 21, 425.
Kähler, E.. (1962). Rendiconti di Matematica 21, 425.
Klein, F. (1872) ver Cartan III.2 , p 1279
Levi-Civita, T. (1917). Rendiconti Di Palermo 42, 173.
Mead, C. A. (2000). Collective Electrodynamics (MIT Press, Cambridge, Massachusetts).
Muraskin, M. (1995) “Mathematical Aesthetic Principles/ Nonintegrable Systems”, World Scientific, Singapore.
Pais, A. (1982), “Subtle is the Lord…”, Oxford Univ. Press, N.Y., p.346.
Parra, J. (1992). On Dirac and Dirac-Darwin-Hestenes Equations in Clifford Algebras and Their applications in Mathematical Physics, A. Micali, R. Boudet and J. Helmstetter, eds., Kluwer Academic Publishers, Dordrecht, pp. 463-477.
Pommaret, J. F. (1988). Lie Pseudogroups and Mechanics (Gordon and Breach, N.Y.).
Rembielinski, J. (1980). Physics Letters
Ricci and Levi-Civita (1901)
Riemann, B. (1854). Über die Hypothesen, welche der Geometrie zu Grunde liegen, Habilitationschrift, Göttingen. Reprinted (1953): Paper XIII in Collected Works of Bernhard Riemann. Weber H. (ed.), Dover, New York.
Riemann B. (1861). Comentatio Mathematica. Collected Works: Paper XXII.
Ringermacher, H. (1994) “An Electrodynamic Connection”, Class. Quantum Grav., Vol 11, pp. 2383-2394.
Sharpe, R. W. (1996) “Differential Geometry: Cartan’s generalization of Klein’s Erlangen Program”(Springer, New York)
Salzer, H. E. (1973) “Two Letters from Einstein Concerning his Distant Parallelism Field Theory” Archives History Exact Sciences, Vol. 12, pp. 88-96.
Jose Vargas and Doug Torr will make every effort to respond to questions on their papers and the subject of Cartan-Einstein-Kähler Unification.