anti-CRF functions, see (6.18) for the definition, have some interesting properties,
cf. Theorem 6.20
In Chapter 7 we study infinitesimal conformal automorphisms of QC struc-
tures (QC-vector fields) and show that they depend on three functions satisfying
some differential conditions thus establishing a ’3-hamiltonian’ form of the QC-
vector fields (Proposition 7.8). The formula becomes very simple expression on
a 3-Sasakian manifolds (Corollary 7.9). We characterize the vanishing of the tor-
sion of Biquard connection in terms of the existence of three vertical vector fields
whose flow preserves the metric and the quaternionic structure. Among them, 3-
Sasakian manifolds are exactly those admitting three transversal QC-vector fields
(Theorem 7.10, Corollary 7.13).
In the last section we complete the proof of our main result Theorem 1.2.
Notation 1.4. a) Let us note explicitly, that in this paper for a one form θ
we use
dθ(X, Y ) = Xθ(Y ) Y θ(X) θ([X, Y ]).
b) We shall denote with ∇h the horizontal gradient of the function h, see (5.1),
while dh means as usual the differential of the function h.
c) The triple {i, j, k} will denote a cyclic permutation of {1, 2, 3}, unless it is ex-
plicitly stated otherwise.
Previous Page Next Page