Cayley form, comass, and triality isomorphisms

  title={Cayley form, comass, and triality isomorphisms},
  author={Mikhail G. Katz and Steven Shnider},
  journal={Israel Journal of Mathematics},
Following an idea of Dadok, Harvey and Morgan, we apply the triality property of Spin(8) to calculate the comass of self-dual 4-forms on ℝ8. In particular, we prove that the Cayley form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.5)) is SO(8)-conjugate to the Cayley form. We also use triality to prove that the stabilizer in SO(8) of the Cayley form is Spin(7). The results have applications in systolic geometry, calibrated geometry, and Spin(7… 
6 Citations
Hyperellipticity and systoles of Klein surfaces
Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard’s short loops on such
Cohomological dimension, self-linking, and systolic geometry
Given a closed manifold M, we prove the upper bound of $${1 \over 2}(\dim M + {\rm{cd}}({{\rm{\pi }}_1}M))$$ for the number of systolic factors in a curvature-free lower bound for the total volume of
We prove a systolic inequality for a {relative systole of a {essential 2{complexX, where : 1(X)! G is a homomor- phism to a nitely presented group G. Thus, we show that univer- sally for any
Loewner’s Torus Inequality with Isosystolic Defect
AbstractWe show that Bonnesen’s isoperimetric defect has a systolic analog for Loewner’s torus inequality. The isosystolic defect is expressed in terms of the probabilistic variance of the conformal
Bi-Lipschitz approximation by finite-dimensional imbeddings
Gromov’s celebrated systolic inequality from ’83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding
Hurwitz quaternion order and arithmetic Riemann surfaces
We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry.


E_7, Wirtinger inequalities, Cayley 4-form, and homotopy
We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying
Geometric structures on G2 and Spin (7)-manifolds
This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and
Small values of the Lusternik-Schnirelmann category for manifolds
We prove that manifolds of Lusternik–Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna by generalizing their
Metrics with exceptional holonomy
It is proved that there exist metrics with holonomy G2 and Spin(7), thus settling the remaining cases in Berger's list of possible holonomy groups. We first reformulate the "holonomy H" condition as
Compact Manifolds with Special Holonomy
The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kahler geometry. Then the Calabi conjecture is proved and used to deduce the existence
Counterexamples to isosystolic inequalities
We explore M. Gromov's counterexamples to systolic inequalities. Does the manifoldS2 ×S2 admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit
An Optimal Loewner-type Systolic Inequality and Harmonic One-forms of Constant Norm
We present a new optimal systolic inequality for a closed Riemannian manifold X, which generalizes a number of earlier inequalities, including that of C. Loewner. We characterize the boundary case of
Boundary case of equality in optimal Loewner-type inequalities
We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, G), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number.