If h and k are symmetric (0, 2)-tensors, then the product is defined via:[1]
where the Xj are tangent vectors and is the matrix determinant. Note that , as it is clear from the second expression.
With respect to a basis of the tangent space, it takes the compact form
where denotes the total antisymmetrisation symbol.
The Kulkarni–Nomizu product is a special case of the product in the graded algebra
where, on simple elements,
( denotes the symmetric product).
Properties[]
The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor.[2] For instance, on space forms (i.e. spaces of constant sectional curvature) and two-dimensional smooth Riemannian manifolds, the Riemann curvature tensor has a simple expression in terms of the Kulkarni–Nomizu product of the metric with itself; namely, if we denote by
When there is a metric tensorg, the Kulkarni–Nomizu product of g with itself is the identity endomorphism of the space of 2-forms, Ω2(M), under the identification (using the metric) of the endomorphism ring End(Ω2(M)) with the tensor product Ω2(M) ⊗ Ω2(M).
A Riemannian manifold has constant sectional curvaturek if and only if the Riemann tensor has the form
^Some authors include an overall factor 1/2 in the definition.
^A (0, 4)-tensor that satisfies the skew-symmetry property, the interchange symmetry property and the first (algebraic) Bianchi identity (see symmetries and identities of the Riemann curvature) is called an algebraic curvature tensor.
References[]
Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN978-3-540-15279-8.
Gallot, S., Hullin, D., and Lafontaine, J. (1990). Riemannian Geometry. Springer-Verlag.CS1 maint: multiple names: authors list (link)