3 5 6 D O C . 2 1 6 R I E M A N N I A N G E O M E T R Y Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1928): 217–221. Presented 7 June 1928, published 10 July 1928. A draft is also available [4 026]. [1] Submitted by Max Planck (see Doc. 218). [2] For a discussion of Einstein’s approach to unified field theory based on teleparallel geometry that begins with this paper and will keep him occupied for the next few years, see the Introduction, sec. V. For historical discussion of Einstein’s teleparallel approach to unified field theory, see also Sauer 2006, 2014. [3] The footnote is missing in the draft. [4] In the draft, this equation replaces . [5] In the draft, the half-sentence “die eines kovarianten” is deleted between “kontravarianten” and “Vektor.” [6] The n-Bein at the end of this sentence should be , as in the draft. [7] In the draft, the following sentence is deleted: “Die ’sind der Vektorcharakter der … dagegen nicht Vektorkomponenten.” The following sentence then replaces “Den Vektorcharakter der erhält man aus (4) und (5)”. [8] In the draft, “parallel” is not in parentheses. [9] Interpreted in modern terms, equation (3) reduces a global symmetry to a global symmetry, which corresponds to the group of rotations in a four-dimensional space-time. [10] In the first equation of (7a), it should be rather than , as in the draft. [11] For earlier discussions of this connection see, e.g., Weitzenböck 1921 and Eisenhart 1925. Teleparallel geometry featuring this connection is a special case of a type of geometry discussed also by Cartan in a series of papers in 1922 and 1923 (see the correspondence between Einstein and Weitzenböck and with Cartan later in this volume). The current document is the first publication in which Einstein explores the consequences of a non-symmetric connection, though he had considered the option before in Vol. 13, Doc. 418. [12] In the draft, the following words between “und” and “ist unsymmetrisch” are crossed out: “hat nichts zu thun mit dem Gesetz der Metrik.” [13] If a metric has been introduced, then one can explain the fact that the components of the Riemann tensor corresponding to the connection vanish identically by noting that the curvature con- tributions of the Levi-Civita connection corresponding to the metric (3) and that of the contorsion ten- sor (see note 16) cancel. Eisenhart 1927 showed that the vanishing of the affine Riemann tensor is a necessary and sufficient condition for the existence of n linearly independent n-beins in an n-dimen- sional space(time). The last is missing a superscript . [14] This is often called the Levi-Civita connection, the unique affine, symmetric connection com- patible with the metric previously introduced via equation (3). [15] This tensor is now called the contorsion tensor it is a linear combination of components of the torsion tensor described in the following note. [16] This tensor is now called the torsion tensor. It was first introduced (and thus named) in Cartan 1922. After equation (10), the draft contains the following crossed-out paragraph: “welcher gemäss (7a) nur erste Differentialquotienten aus den h enthält. ist sowohl Koordinatentransformationen als auch Drehungstransformationen gegenüber kovariant. Aus (10) folgen die beiden Kovarianten also . . . . (11) und . . . . (12) sowie die beiden zugehörigen Invarianten . . . . (13) . . . . (14) A2 A2 a hhdxdx = = h a h a h a h a h a GL 4 SO3 1 dx dx V = U = I 1 g = I 2 g =