Riemannian Holonomy? Jacob Gross Communicated by Cesar E

THEGRADUATESTUDENTSECTION

WHATIS… Riemannian holonomy? Jacob Gross Communicated by Cesar E. Silva

Berger’s classification of Riemannian holonomy groups is If 푀 is path-connected then Hol푝(푔) is, up to conju- a strong organizing principle in differential geometry. It gacy, independent of 푝. One should always regard the tells us about exceptional geometric structures, existing holonomy group as coming with a natural representation: only in certain dimensions, that occupy central roles in the inclusion Hol푝(푔) → GL(푇푝푀). In 1926 Élie Cartan ob- physics, and generally serves as a road map for some served that the holonomy group acts reducibly if and only research trends in algebraic and symplectic geometry. if the metric is (locally) a product metric. This reduces the To each Riemannian manifold (푀, 푔) there are associ- problem of classifying holonomy groups of Riemannian ated parallel transport maps 푃훾 that move vectors along manifolds to the problem of classifying holonomy groups a path 훾 ∶ 퐼 → 푀 such that the motion looks parallel of irreducible Riemannian manifolds. Cartan also wrote from the metric’s point of view. Fixing a point 푝 in 푀 the down all the holonomy groups of so-called ‘symmetric spaces.’ A symmetric space is a Riemannian manifold 푀 holonomy group is the group, written Hol푝(푔), of parallel transport maps around loops based at 푝, as in Figure 1. such that, at each 푝 ∈ 푀, the geodesic reflection 푠푝 is an 푛 푛 N isometry. Euclidean spaces ℝ , spheres 푆 , and hyperbolic spaces H푛 are all examples of symmetric spaces. For (simply-connected) irreducible nonsymmetric Riem- manian manifolds, Marcel Berger wrote down a list of all possible holonomy groups. Theorem (Berger, 1955). Let (푀, 푔) be a simply-connected, irreducible, nonsymmetric Riemannian manifold. Let 푛 = dim 푀. Then the holonomy group Hol(푔) of (푀, 푔) is either a • SO(푛), A B • U(푚) with 푛 = 2푚 and 푚 ≥ 2, • SU(푚) with 푛 = 2푚 and 푚 ≥ 2, • Sp(푚) with 푛 = 4푚 and 푚 ≥ 2, • Sp(푚)Sp(1) with 푛 = 4푚 and 푚 ≥ 2, • 퐺2 with 푛 = 7, S • Spin(7) with 푛 = 8, or Figure 1. The holonomy group results from • Spin(9) with 푛 = 16. parallel-translating a vector around all loops from a As it turns out, there are no irreducible nonsymmetric fixed point A. On the round sphere, it includes all Riemannian manifolds with holonomy group equal to rotations, but on some manifolds, it is a smaller Spin(9). In 1968 Alexeevsky eliminated Spin(9) from this subgroup. list. All other entries on Berger’s list, however, do occur as the holonomy group of some irreducible nonsymmetric Riemannian manifold, although it took some time to Jacob Gross is a PhD student at Oxford University. His email realize this. address is [email protected]. Manifolds with holonomy contained in U(푚) are called For permission to reprint this article, please contact: Kähler, manifolds with holonomy contained in SU(푚) are [email protected]. called Calabi–Yau, and those with holonomy contained in DOI: http://dx.doi.org/10.1090/noti1701 Sp(푚) are called hyperkähler.

August 2018 Notices of the AMS 795 THEGRADUATESTUDENTSECTION

Berger’s list, in a sense, echoes the classification of real normed division algebras. ACKNOWLEDGMENTS. The author would like to thank Dominic Joyce and a referee for providing useful Theorem (Dickson). There are exactly four real normed comments, for offering editorial suggestions, and for division algebras: the real numbers ℝ, the complex num- assisting with revisions of this article. The author also bers ℂ, the quaternions ℍ, and the octonions 핆. acknowledges support from the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Each group in Physics during the time that he wrote this piece. Berger’s list is a Manifolds with group whose ele- special holonomy ments are automor- References phisms/isometries of [1] R. Bryant, Metrics with Exceptional Holonomy, Ann. of Math. are important in a vector space over 126 (1987), 525–576. MR916718 some real divison al- [2] A. Corti, M. Haskins, J. Nordstrom, and T. Pacini, physics. gebra. For example G2-manifolds and associative submanifolds via semi-Fano 3- SO(푚) (resp. SU(푚)) folds, Duke Math. J. 164 (2015) 1971–2092, arXiv:1207.4470. is a group of automor- MR3369307 phisms of ℝ푚 (resp. ℂ푚). In this sense, Sp(푚) and [3] D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry, Oxford University Press, 2007. MR2292510 Sp(푚)Sp(1) holonomies are quaternionic geometries, [4] A. Kovalev, Twisted connected sums and special Riemann- while 퐺2 holonomy and Spin(7) holonomy are octo- ian holonomy, J. Reine Angew. Math. 565 (2003), 125–160. nionic geometries; 퐺2 is the group of automorphisms of MR2024648 핆 and Spin(7) is the group of isometries of the octo- [5] S. Salamon, Riemannian geometry and holonomy groups, nions 핆 generated by left multiplication by unit length Pitman Research Notes in Mathematics Series, vol. 201, imaginary octonions. Longman Scientific & Technical, Harlow, 1989. MR1004008 Manifolds with special holonomy are important in physics. One reason is that a so-called “parallel spinor Image Credits field” is required for the equations of supersymmetry Figure 1 by Fred the Oyster [CC BY-SA 4.0 (https:// to work. On a general Riemannian manifold, the parallel creativecommons.org/licenses/by-sa/4.0)], via Wiki- tensors determine the holonomy group. On a spin mani- media Commons. fold, the holonomy group determines the parallel spinors. Author photo by Michael Williams. For this reason, manifolds with special holonomy groups (especially SU(3) and 퐺2) can be useful in theoretical physics. ABOUT THE AUTHOR Manifolds with holonomy SU(3), Calabi–Yau 3- folds, form so-called “string compactifications” in When not doing mathematics, ten-dimensional supersymmetric string theories. This Jacob Gross trains in acrobatics means that, in such a string theory, the universe is locally and in break dancing. modeled on ℝ(1,3) × 푋, where ℝ(1,3) denotes Minkowski spacetime and 푋 is a Calabi–Yau manifold of 6 real dimensions. In 푀-theory, the universe is supposed to have (1,3) 11 dimensions and to be locally modelled on ℝ × 푋, Jacob Gross where 푋 is a compact (singular) seven-dimensional manifold with holonomy 퐺2. 퐺2 and Spin(7) manifolds are rather unlike Calabi–Yau manifolds, however, in that it is notoriously hard to write down examples. To illustrate the difficulty: finding a metric with holonomy 퐺2 amounts to solving a simulta- neous system of forty-nine nonlinear PDEs. Bryant wrote down the first metrics with holonomy 퐺2 and with holonomy Spin(7) in 1987. These metrics were not complete. Later Bryant and Salamon constructed complete noncom- pact examples of manifolds with holonomomy 퐺2 and Spin(7). In 1993 Joyce constructed the first examples of compact 7-manifolds with holonomy 퐺2 and of compact 8-manifolds with holonomy Spin(7). Since then other examples of compact manifolds with holonomy 퐺2 have been constructed by Corti–Haskins–Pacini–Nördstrom, Joyce–Karigiannis, and Kovalev.

796 Notices of the AMS Volume 65, Number 7