Schouten tensor
Encyclopedia
In Riemannian geometry
, the Schouten tensor is a second-order tensor which is introduced by Jan Arnoldus Schouten
. It is defined by, for n ≥ 3 dimensions,
where Ric is the Ricci tensor, R is the scalar curvature
, g is the Riemannian metric and n is the dimension of the manifold.
The Weyl tensor
equals the Riemann curvature tensor
minus the Kulkarni–Nomizu product of the Schouten tensor with the metric.
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, the Schouten tensor is a second-order tensor which is introduced by Jan Arnoldus Schouten
Jan Arnoldus Schouten
Jan Arnoldus Schouten was a Dutch mathematician. He was an important contributor to the development of tensor calculus and was one of the founders of the Mathematisch Centrum in Amsterdam....
. It is defined by, for n ≥ 3 dimensions,
where Ric is the Ricci tensor, R is the scalar curvature
Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
, g is the Riemannian metric and n is the dimension of the manifold.
The Weyl tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
equals the Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
minus the Kulkarni–Nomizu product of the Schouten tensor with the metric.