[Paper Review] Distinguished dimensions for special Riemannian geometries

Paweł Nurowski|ArXiv.org|Jan 2, 2006

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper identifies special Riemannian geometries with structure groups $H_k = \mathrm{SO}(3), \mathrm{SU}(3), \mathrm{Sp}(3), \mathbf{F}_4$ in dimensions $n_k = 5, 8, 14, 26$, arising from ternary symmetric forms invariant under these groups. It establishes necessary and sufficient conditions for such geometries to admit a characteristic connection with totally skew-symmetric torsion, and constructs explicit examples in dimension 8 with nonvanishing torsion satisfying Einstein equations under both Levi-Civita and characteristic connections.

ABSTRACT

The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan's works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions n_k=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group to the subgroups H_k\subset SO(n_k), with H_1=SO(3), H_2=SU(3), H_4=Sp(3) and H_8=F_4. This enables studies of special Riemannian geometries with structure groups H_k in dimensions n_k. The neccessary and sufficient conditions for the H_k geometries to admit the characteristic connection are given. As an illustration nontrivial examples of SU(3) geometries in dimension 8 admitting characteristic connection are provided. Among them there are examples having nonvanishing torsion and satisfying Einstein equations with respect to either the Levi-Civita or the characteristic connections.

Motivation & Objective

• To identify dimensions and geometric structures admitting a ternary symmetric form that reduces the orthogonal group to special subgroups $H_k$.
• To determine the necessary and sufficient conditions for the existence of a characteristic connection with totally skew-symmetric torsion in $H_k$-geometries.
• To construct nontrivial examples of $\mathrm{SU}(3)$-geometries in dimension 8 with nonvanishing torsion that satisfy Einstein equations under both Levi-Civita and characteristic connections.
• To explore the implications of the 'magic square' of Lie groups for constructing ten additional classes of special Riemannian geometries in higher dimensions.
• To analyze the $\mathrm{SU}(2)\times\mathrm{SU}(2)$-structure in dimension 8 as a special case, identifying the tensor that reduces $\mathrm{SO}(8)$ to this subgroup.

Proposed method

• Utilizes the algebraic structure of ternary symmetric forms invariant under $H_k$ subgroups of $\mathrm{SO}(n_k)$, derived from Cartan's theory of isoparametric hypersurfaces in spheres.
• Applies the nearly integrability condition $\nabla^{\mathrm{LC}}_v \Upsilon(v,v,v) \equiv 0$ to define a unique metric $\mathfrak{so}(3)$-valued characteristic connection $\nabla^T$ with totally skew-symmetric torsion.
• Employs representation theory of $\mathrm{SU}(3)$, $\mathrm{Sp}(3)$, and $\mathbf{F}_4$ to analyze the decomposition of the tangent space and the action of the structure group.
• Uses the 'magic square' construction to predict additional special Riemannian geometries with structure groups $\mathbf{U}(3)$, $\mathbf{S}({\rm U}(3)\times{\rm U}(3))$, $\mathbf{U}(6)$, $\mathbf{E}_6\times\mathrm{SO}(2)$, $\mathbf{Sp}(3)\times\mathrm{SU}(2)$, $\mathbf{SU}(6)\times\mathrm{SU}(2)$, $\mathbf{SO}(12)\times\mathrm{SU}(2)$, and $\mathbf{E}_7\times\mathrm{SU}(2)$ in dimensions 12, 18, 30, 54, 28, 40, 64, and 112.
• Constructs explicit examples in dimension 8 using a 3-tensor $\Upsilon$ defined via the determinant of a $3\times3$ real symmetric trace-free matrix, and verifies the existence of characteristic connections.
• Analyzes the $\mathrm{SU}(2)\times\mathrm{SU}(2)$-structure in $\mathrm{SO}(8)$ by identifying the tensor that reduces the structure group to this product, laying groundwork for further study.

Experimental results

Research questions

• RQ1 In which dimensions $n$ does there exist a symmetric, trace-free ternary tensor $\Upsilon_{ijk}$ satisfying the algebraic identity $\Upsilon_{jki}\Upsilon_{lmi} + \Upsilon_{lji}\Upsilon_{kmi} + \Upsilon_{kli}\Upsilon_{jmi} = g_{jk}g_{lm} + g_{lj}g_{km} + g_{kl}g_{jm}$?
• RQ2 What are the necessary and sufficient conditions for an $H_k$-structure on a Riemannian manifold to admit a characteristic connection with totally skew-symmetric torsion?
• RQ3 Can nontrivial examples of $\mathrm{SU}(3)$-geometries in dimension 8 be constructed that admit a characteristic connection with nonvanishing torsion and satisfy the Einstein equations under both the Levi-Civita and characteristic connections?
• RQ4 How do the 'magic square' Lie groups suggest the existence of ten additional classes of special Riemannian geometries, and what are their respective dimensions and structure groups?
• RQ5 What tensor structure reduces $\mathrm{SO}(8)$ to $\mathrm{SU}(2)\times\mathrm{SU}(2)$, and what are the implications for the geometry of 8-manifolds with such a structure?

Key findings

• The only dimensions $n_k = 3k+2$ for $k=1,2,4,8$ admit a ternary symmetric form $\Upsilon$ satisfying the required algebraic identity, reducing $\mathrm{SO}(n_k)$ to $H_k = \mathrm{SO}(3), \mathrm{SU}(3), \mathrm{Sp}(3), \mathbf{F}_4$.
• For $H_k$-geometries, the existence of a characteristic connection with totally skew-symmetric torsion is equivalent to the nearly integrability condition $\nabla^{\mathrm{LC}}_v \Upsilon(v,v,v) \equiv 0$.
• Explicit examples of $\mathrm{SU}(3)$-geometries in dimension 8 are constructed with 11-dimensional symmetry group and torsion in the $\mathbf{27}$-dimensional irreducible representation of $\mathrm{SU}(3)$, satisfying the Einstein equations with respect to the characteristic connection.
• In dimension 8, examples exist with 9-dimensional symmetry group and vectorial torsion, also satisfying the Einstein equations under the characteristic connection.
• The $\mathrm{SU}(2)\times\mathrm{SU}(2)$-structure in $\mathrm{SO}(8)$ is characterized by a specific tensor derived from the determinant of a $3\times3$ symmetric matrix, reducing the structure group to this product.
• The 'magic square' construction predicts ten additional classes of special Riemannian geometries in dimensions 12, 18, 30, 54, 28, 40, 64, and 112 with structure groups $\mathbf{U}(3)$, $\mathbf{S}({\rm U}(3)\times{\rm U}(3))$, $\mathbf{U}(6)$, $\mathbf{E}_6\times\mathrm{SO}(2)$, $\mathbf{Sp}(3)\times\mathrm{SU}(2)$, $\mathbf{SU}(6)\times\mathrm{SU}(2)$, $\mathbf{SO}(12)\times\mathrm{SU}(2)$, and $\mathbf{E}_7\times\mathrm{SU}(2)$.