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Given a Riemannian metric g, the scalar curvature Scal is defined as the trace of the Ricci curvature tensor with respect to the metric: [1] = . The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace.
(For example, the three-dimensional torus is such a manifold.) However, Kazdan and Warner proved that if M does admit some metric with strictly positive scalar curvature, then any smooth function ƒ is the scalar curvature of some Riemannian metric.
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). [1] [page needed] On the other hand, since a sphere of radius R has constant positive curvature R −2 and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar ...
Here is the Ricci curvature tensor and represents the Lie derivative. If there exists a function f : M → R {\displaystyle f:M\rightarrow \mathbb {R} } such that V = ∇ f {\displaystyle V=\nabla f} we call ( M , g ) {\displaystyle (M,g)} a gradient Ricci soliton and the soliton equation becomes
For example, this formula ... where is the scalar curvature, defined in ... positive Ricci curvature of a Riemannian manifold has strong topological consequences, ...
A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. An example of negatively curved space is hyperbolic geometry (see also: non-positive curvature). A space or space-time with zero curvature is called flat.
These facts were used by LeBrun and Salamon [11] to prove that, up to isometry and rescaling, there are only finitely many positive-scalar-curvature compact quaternion-Kähler manifolds in any given dimension. This same paper also shows that any such manifold is actually a symmetric space unless its second homology is a finite group with non ...
The Weyl tensor is invariant with respect to a conformal change of metric: if two metrics are related as ~ = for some positive scalar function , then ~ = . In dimensions 2 and 3 the Weyl tensor vanishes, but in 4 or more dimensions the Weyl tensor can be non-zero.