Adv. Appl. Cliﬀord Algebras 27 (2017), 2585–2605
2017 Springer International Publishing
published online March 3, 2017
Applied Cliﬀord Algebras
Derivations and Linear Connections on
Cliﬀord Vector Bundles
Communicated by RafalAblamowicz
Abstract. The goal of this article is to introduce a concept of Cliﬀord
structures on vector bundles as natural extensions of the standard com-
plex and quaternionic structures, and to determine the derivations and
linear connections on smooth Cliﬀord vector bundles compatible with
their Cliﬀord structures. The basic object used to get such descriptions
is an involution on the space of derivations of a Cliﬀord vector bundle
explicitly deﬁned in terms of the speciﬁc Cliﬀord structure. That involu-
tion is actually derived from an operation called the Cliﬀord conjugation
relative to a Cliﬀord structure, which is deﬁned in a purely algebraic set-
ting as an involution on the space of derivations of a Euclidean Cliﬀord
algebra. Its deﬁnition essentially relies on the use and a complete de-
scription of the geometric concept of tangent Cliﬀord structures of a
Euclidean Cliﬀord algebra.
Mathematics Subject Classiﬁcation. 15A66, 53C07, 53C15, 53C27.
Keywords. Cliﬀord structures, Derivations, Linear connections.
To motivate in part the main theme of this article, we will begin with a
quick account of the deﬁnitions of Laplace and Dirac operators on smooth
inner product vector bundles. For more details and excellent introductions to
spin geometry, the speciﬁc area in diﬀerential geometry concerned with the
study of Dirac operators in a diﬀerential geometric setting, we refer to the
Let M be a Riemann manifold of dimension m ≥ 1, and let TM and
M be the tangent and cotangent bundles of M. Further, let G(M )and
C(M)betheGrassmann and Cliﬀord algebra bundles of M, respectively.
Their ﬁbers at x ∈ M equal the graded exterior algebra Λ