Exceptional Symmetric Spaces Research Articles

Totally geodesic submanifolds in exceptional symmetric spaces

We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.

The index conjecture for symmetric spaces

Abstract In 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank ≤ 2 {\leq 2} , but for higher rank it was unclear how to tackle the problem. In [J. Berndt, S. Console and C. E. Olmos, Submanifolds and holonomy, 2nd ed., Monogr. Res. Notes Math., CRC Press, Boca Raton 2016], [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], [J. Berndt and C. Olmos, The index of compact simple Lie groups, Bull. Lond. Math. Soc. 49 2017, 5, 903–907], [J. Berndt and C. Olmos, On the index of symmetric spaces, J. reine angew. Math. 737 2018, 33–48], [J. Berndt, C. Olmos and J. S. Rodríguez, The index of exceptional symmetric spaces, Rev. Mat. Iberoam., to appear] we developed several approaches to this problem, which allowed us to calculate the index for many symmetric spaces. Our systematic approach led to a conjecture, formulated first in [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], for how to calculate the index. The purpose of this paper is to verify the conjecture.

The absolute continuity of convolution products of orbital measures in exceptional symmetric spaces

Let G be a non-compact group, K the compact subgroup fixed by a Cartan involution and assume G / K is an exceptional, symmetric space, one of Cartan type E, F or G. We find the minimal integer, L(G), such that any convolution product of L(G) continuous, K-bi-invariant measures on G is absolutely continuous with respect to Haar measure. Further, any product of L(G) double cosets has non-empty interior. The number L(G) is either 2 or 3, depending on the Cartan type, and in most cases is strictly less than the rank of G.

Harmonic maps into the exceptional symmetric space G2 /SO(4)

We show that a harmonic map from a Riemann surface into the exceptional symmetric space G 2 / SO ( 4 ) has a J 2 -holomorphic twistor lift into one of the three flag manifolds of G 2 if and only if it is ‘nilconformal’, that is, has nilpotent derivative. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into G 2 / SO ( 4 ) that are not of finite uniton number, and that have lifts into any of the three twistor spaces. Harmonic maps of finite uniton number are all nilconformal; for such maps, we show that our lifts can be constructed explicitly from extended solutions.

Polar supermultiplets, hermitian symmetric spaces and hyperkähler metrics

We address the construction of four-dimensional N=2 supersymmetric nonlinear sigma models on tangent bundles of arbitrary Hermitian symmetric spaces starting from projective superspace. Using a systematic way of solving the (infinite number of) auxiliary field equations along with the requirement of supersymmetry, we are able to derive a closed form for the Lagrangian on the tangent bundle and to dualize it to give the hyperkahler potential on the cotangent bundle. As an application, the case of the exceptional symmetric space E_6/SO(10) \times U(1) is explicitly worked out for the first time.

Generalized projective geometries: general theory and equivalence with Jordan structures

In this work we introduce generalized projective geometries which are a natural generalization of projective geometries over a field or ring K but also of other important geo- metries such as Grassmannian, Lagrangian or conformal geometry (see (3)). We also introduce the corresponding generalized polar geometries and associate to such a geometry a symmetric space over K. In the finite-dimensional case over K ¼ R, all classical and many exceptional symmetric spaces are obtained in this way. We prove that generalized projective and polar geometries are essentially equivalent to Jordan algebraic structures, namely to Jordan pairs, respectively to Jordan triple systems over K which are obtained as a linearized tangent version of the geometries in a similar way as a Lie group is linearized by its Lie algebra. In contrast to the case of Lie theory, the construction of the ''Jordan functor'' works equally well over gen- eral base rings and in arbitrary dimension.