Bochner formulae for orthogonal G-structures on compact manifolds
by
Luis Hernández-Lamoneda
Centro de Investigación en Matemáticas (CIMAT), Mexico
Coauthors: Gil Bor (CIMAT)

Oral Communication

Let G subset O_n be one of the groups U_ⁿ/2, SU_ⁿ/2, Sp_ⁿ/4·Sp₁, G₂ (n=7) or Spin₇ (n=8); in all cases, G is the stabilizer (in O_n) of a certain p-form \phi (p=2, ⁿ/₂, 4, 3 and 4 respectively). If M is an n-dimensional manifold with a G-structure then its intrinsic torsion can be identified with the covariant derivative Ñ\phi of the associated p-form on M (denoted by \phi as well). Thus, the (local) holonomy of the Levi-Civita connection on TM is contained in G if and only if \phi is parallel, Ñ\phi = 0. If this does not happen, one can decompose Ñ\phi into G-irreducible components, Ñ\phi = (Ñ\phi)₁\oplus(Ñ\phi)₂\oplus ... . These components carry interesting geometric information about the G-structure. For example, for G=U_m, A. Gray and L. Hervella showed that Ñ\phi has 4 irreducible components, the sum of two of them measuring the intrinsic torsion of the associated almost-complex structure (the Nijenhuis tensor), the sum of certain three of them representing the intrinsic torsion of the corresponding almost symplectic structure (the exterior derivative of the Kähler form), etc.

Using the classical Bochner-Weitzenbock formulas for the laplacian on p-forms and certain assumptions on G (holding for all cases above except for SU_m), we get for a compact manifold M an integral formula of the form

å i  ci ó õ M  |(Ñ\phi)i|2= ó õ M  s( R),

where s is a certain G-invariant of the curvature and the c_i are constants that can be explicitly determined by representation theoretic calculations.

The formulas are very general (they apply to any G-structure, for the groups considered, on any compact manifold), and yield several interesting applications.

Two such applications are the following:

1. Let G/K = X=X₁×X₂× ... ×X_k be a symmetric space, where the X_j are its irreducible components. Let M be a compact riemannian quotient of X, and let J be a compatible almost-hermitian structure on M.

   • If X is of non-compact type and J is integrable; or
   • X is of compact type and J is symplectic;
   Then X is Hermitian-symmetric and J descends from one of the 2^k G-invariant Kähler structures on X.

   This extends a result of P. Gauduchon.

2. Let M be a compact quaternionic hermitian manifold. If K_\C <= 0 then M is quaternionic-Kähler. Moreover, if the sectional curvature of M is negative and pointwise 1/4-pinched then M is a quotient of quaternionic-hyperbolic space.

   Quaternionic hermitian is a class of Sp_n·Sp₁-structure that has the property that its twistor space is complex. Quaternionic Kähler means that the structure is parallel. K_\C means the complexified sectional curvature. The last part of the statement follows from the first using a result of S.K. Yeung.

Date received: April 4, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-23.