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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.
The scalar curvature is the total trace of the Riemannian curvature tensor, a smooth function on the manifold (,), and in the Kähler case the condition that the scalar curvature is constant admits a transformation into an equation similar to the complex Monge–Ampere equation of the Kähler–Einstein setting.
The only regular (of class C 2) closed surfaces in R 3 with constant positive Gaussian curvature are spheres. [2] If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature. [3]
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
where Δ is the Laplace-Beltrami operator (of negative spectrum), and R is the scalar curvature. This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric. If n ≥ 3 and g is a metric and u is a smooth, positive function, then the conformal metric
M is three-dimensional and g 0 has positive Ricci curvature; M has dimension greater than three and the product metric on (M, g 0) × ℝ has positive isotropic curvature; the normalized Ricci flow with initial data g 0 exists for all positive time and converges smoothly, as t goes to infinity, to a metric of constant curvature.
The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiation) and Lie bracket [,] by the following formula: (,) = [,].
The mean curvature is an extrinsic measure of curvature equal to half the sum of the principal curvatures, k 1 + k 2 / 2 . It has a dimension of length −1. Mean curvature is closely related to the first variation of surface area. In particular, a minimal surface such as a soap film has mean curvature zero and a soap bubble has