Adv. Appl. Cliﬀord Algebras 27 (2017), 2333–2343
2017 Springer International Publishing
published online March 21, 2017
Applied Cliﬀord Algebras
Einstein Metrics Induced by Natural
,S¸emsi Eken Meri¸c and Erol Kılı¸c
Communicated by Vladim´ır Soucek
Abstract. Cliﬀord algebras are used in theoretical physics and in partic-
ular, in the general theory of relativity, where Einstein’s equations are
rewritten in Girard (Adv Appl Cliﬀord Algebras 9(2):225–230, 1999)
within a Cliﬀord algebra. Let M be a manifold with a torsion-free con-
nection which induces on its cotangent bundle T
M, a semi-Riemannian
metric ¯g, called the natural Riemann extension, Kowalski and Sekizawa
(Publ Math Debrecen 78:709–721, 2011). The main result of the present
paper gives a necessary and suﬃcient condition for ¯g restricted to certain
hypersurfaces of T
M to be Einstein.
Mathematics Subject Classiﬁcation. 53C25, 58B20, 15A66.
Keywords. Cliﬀord algebra, Cotangent Bundle, Einstein manifold,
Natural Riemann Extension.
Ricci curvature plays a fundamental role in general relativity, especially in
the Einstein ﬁeld equations. The existence or the non-existence of Einstein
metrics on a manifold is related to some Cliﬀord algebras, as shown in many
papers. A few examples are pointed out in what follows.
Throughout this note, by a positive Einstein metric we mean an Einstein
metric with positive scalar curvature.
The use of Weitzenb¨ock formula for Dirac operators yields to several
examples of manifolds of dimension ≥5, which do not admit any positive
Einstein metric. Moreover, a K3 surface (from Kodaira’s classiﬁcation) is a
complex surface with vanishing ﬁrst Chern class and no global holomorphic
one-forms. This spin surface admits no metric with positive scalar curvature
Dedicated to the memory of Professor Cristian Ida.