The Ricci tensor is obtained from the Riemann tensor by contracting one pair of indices: Ric(X,Y) = trace(Z ↦ R(Z,X)Y). What geometric information does positive Ricci curvature encode?
AGeodesics curve toward each other, and geodesic balls have smaller volume than Euclidean balls of the same radius
BThe manifold has positive Gaussian curvature at every point
CParallel transport around any loop is the identity
DThe manifold is diffeomorphic to a sphere
Positive Ricci curvature in a direction v means that a thin cone of geodesics emanating from p in directions near v converges — the volume of a small geodesic ball grows slower than in Euclidean space. The Bishop-Gromov volume comparison theorem makes this precise. Positive Ricci curvature does not imply positive Gaussian curvature (that is a stronger condition in dimensions > 2). It also does not force the manifold to be a sphere, although the Bonnet-Myers theorem does force compactness and finite fundamental group.
Question 2 True / False
The scalar curvature R = gⁱʲRicᵢⱼ is the simplest curvature invariant. On a 2-dimensional surface, R equals twice the Gaussian curvature K.
TTrue
FFalse
Answer: True
In two dimensions, the Riemann tensor has only one independent component (up to symmetries), and all curvature notions reduce to the Gaussian curvature K. Specifically, Rᵢⱼₖₗ = K(gᵢₖgⱼₗ - gᵢₗgⱼₖ), the Ricci tensor is Ricᵢⱼ = Kgᵢⱼ, and the scalar curvature is R = 2K. In higher dimensions, scalar curvature carries much less information than the full Riemann tensor — it is the weakest curvature condition.
Question 3 Short Answer
In Einstein's field equations Ric - ½Rg + Λg = 8πT, the left side involves only the Ricci tensor and scalar curvature, not the full Riemann tensor. Why is the full Riemann tensor not needed?
Think about your answer, then reveal below.
Model answer: The Einstein equations govern the relationship between matter (T) and curvature. The Ricci tensor captures how matter sources curve spacetime (specifically, how geodesic balls change volume), while the remaining part of the Riemann tensor — the Weyl tensor — describes gravitational radiation (curvature in vacuum). The Weyl tensor propagates freely via the Bianchi identities and does not need to be specified by the field equations. The Einstein equations determine the Ricci part; the Weyl part is determined by boundary/initial conditions and propagation.
The decomposition Riemann = Ricci part + Weyl part separates curvature into the 'matter-determined' piece and the 'freely propagating' piece. In 3 dimensions, the Weyl tensor vanishes identically, so the Ricci tensor determines the full Riemann tensor and there is no gravitational radiation. In 4+ dimensions, the Weyl tensor is nonzero and carries independent information about the geometry.