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Curvature and the Ricci Tensor

Summary

Curvature measures how far a Riemannian manifold deviates locally from Euclidean space. Out of the Riemann curvature tensor one obtains, by tracing, sectional curvature, the Ricci tensor and the scalar curvature. The Ricci tensor – an averaged notion of curvature – is the right-hand side of Hamilton's Ricci flow.

1. Why Curvature?

A flat plane differs from a sphere because geodesics on the sphere converge while in the plane they remain parallel. Curvature makes this behaviour quantitative. On a Riemannian manifold \((M, g)\) the Levi-Civita connection (Article 01, §6) encodes the failure of covariant derivatives to commute.

2. The Riemann Curvature Tensor

For vector fields \(X, Y, Z\):

\[R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z.\]

\(R\) is tensorial – \(R_p(X,Y)Z\) depends only on the values of \(X,Y,Z\) at \(p\). In index notation:

\[R^l{}_{ijk} = \partial_i \Gamma^l_{jk} - \partial_j \Gamma^l_{ik} + \Gamma^l_{im}\Gamma^m_{jk} - \Gamma^l_{jm}\Gamma^m_{ik}.\]

With lowered index \(R_{lijk} = g_{lm} R^m{}_{ijk}\) the symmetries read

  • antisymmetry: \(R_{lijk} = -R_{iljk}\) and \(R_{lijk} = -R_{likj}\),
  • block symmetry: \(R_{lijk} = R_{jkli}\),
  • first Bianchi identity: \(R_{l[ijk]} = 0\),
  • second Bianchi identity: \(\nabla_{[m} R_{li]jk} = 0\).

These reduce, on an \(n\)-manifold, the number of independent components from \(n^4\) to \(\tfrac{1}{12}n^2(n^2-1)\).

3. Sectional Curvature

For two linearly independent \(u, v \in T_pM\) the sectional curvature of the plane spanned by \(u, v\) is

\[K(u, v) = \frac{g\bigl(R(u,v)v, u\bigr)}{g(u,u)\,g(v,v) - g(u,v)^2}.\]

It generalises the Gaussian curvature of surfaces. Constant sectional curvature characterises the three model geometries:

Sectional curvature Model space (dim. \(n\)) Geometry
\(K \equiv +1\) sphere \(S^n\) spherical
\(K \equiv 0\) Euclidean space \(\mathbb{R}^n\) flat
\(K \equiv -1\) hyperbolic space \(\mathbb{H}^n\) hyperbolic

In dimension 3 constant sectional curvature is not the end of the story – Thurston's classification recognises eight model geometries (see Act 1, Article 05).

4. The Ricci Tensor

Tracing two indices of the curvature tensor produces the Ricci tensor:

\[\mathrm{Ric}_{jk} = R^i{}_{jik} = g^{im}\,R_{mjik}.\]

Geometrically, \(\mathrm{Ric}_p(v,v)\) is the average sectional curvature of all 2-planes containing \(v\). Pictorially: for a small geodesic "trumpet" emanating in direction \(v\), the Ricci tensor measures how the infinitesimal volume element evolves along the geodesics – positive Ricci pulls volumes together, negative Ricci spreads them out.

"The Ricci tensor measures the average way in which the volume element distorts as you move along a geodesic." — John W. Morgan & Gang Tian, Ricci Flow and the Poincaré Conjecture (2007), §1.2

The Ricci tensor is a symmetric \((0,2)\)-tensor field – exactly the shape that, when added to a metric, produces a metric variation. This is what makes it suitable as the right-hand side of an evolution equation for \(g\).

5. Scalar Curvature

A further trace yields the scalar curvature:

\[R = g^{jk}\,\mathrm{Ric}_{jk}.\]

It is a smooth function on \(M\). On the round \(S^n\), \(R = n(n-1)\); on \(\mathbb{R}^n\), \(R = 0\); on \(\mathbb{H}^n\), \(R = -n(n-1)\). In dimension 2, \(R\) equals twice the Gaussian curvature and, via Gauss–Bonnet, completely determines the topology of a closed surface.

6. The Special Role of Dimension 3

In dimension 3 the Riemann curvature tensor and the Ricci tensor have the same number of independent components (six each). This implies a key identity:

In dimension 3 the full curvature tensor \(R\) is algebraically determined by the Ricci tensor and the metric.

This is the very reason Hamilton's Ricci flow can be used for classification in 3D: an evolution equation for \(\mathrm{Ric}\) alone is enough to control the entire geometry. In higher dimensions this reduction fails – the Weyl tensor remains, making the problem substantially harder.

7. Volume and Diameter Comparison

Curvature bounds translate into geometric and topological consequences.

  • Bonnet–Myers: if \(\mathrm{Ric} \ge (n-1) k\, g\) with \(k > 0\), then \(M\) is compact and \(\mathrm{diam}(M) \le \pi/\sqrt{k}\).
  • Bishop–Gromov: a lower bound on \(\mathrm{Ric}\) gives an upper bound on the volume growth of geodesic balls.
  • Cheeger–Gromoll splitting: if \(\mathrm{Ric} \ge 0\) and \(M\) contains a complete geodesic line, then \(M\) splits isometrically as \(N \times \mathbb{R}\).

These comparison theorems are central tools in Perelman's analysis of Ricci flow singularities.

8. Einstein Manifolds

A Riemannian manifold is Einstein if

\[\mathrm{Ric} = \lambda\, g \quad \text{for some } \lambda \in \mathbb{R}.\]

Such metrics are fixed points of the normalised Ricci flow (see Article 03). The round sphere, Euclidean space and hyperbolic space are Einstein, as are symmetric spaces more generally. Finding Einstein metrics on a given manifold is a research field in itself (Yamabe problem).

9. Towards the Flow

With Riemannian metric (Article 01), curvature tensor and Ricci tensor in hand, all building blocks are in place to formulate the evolution equation

\[\frac{\partial g}{\partial t} = -2\,\mathrm{Ric}(g)\]

and to analyse its action. That is the topic of the next article.

Sources

  • John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer (2018), Ch. 7.
  • Manfredo do Carmo, Riemannian Geometry, Birkhäuser (1992), Ch. 4–6.
  • John W. Morgan & Gang Tian, Ricci Flow and the Poincaré Conjecture, AMS Clay Math. Monographs Vol. 3 (2007), §1–2.
  • Peter Petersen, Riemannian Geometry, 3rd ed., Springer (2016), Ch. 3 & 9.

Cross References