Curvature Tensor

LIGO (2015, Nobel 2017) detected gravitational waves from 1.3 billion light-years away -- a solution of Einstein's equations G_{mu nu}=8pi T_{mu nu}, where G is expressed through the Ricci curvature tensor.

• **GR:** Einstein tensor G_{mu nu}=R_{mu nu}-half R g_{mu nu}=8pi T_{mu nu}. Bianchi identity gives nabla^mu G_{mu nu}=0 automatically.
• **Topology:** Gauss-Bonnet is the prototype of the Atiyah-Singer theorem linking geometry and operator spectra.
• **Riemannian ML:** curvature tensor measures non-convexity for optimization on manifolds.

The Riemann Curvature Tensor

Einstein's GR equations G_{mu nu}=8pi T_{mu nu} link the Einstein tensor (expressed through the Ricci tensor and metric) to the energy-momentum tensor. The LIGO detection of gravitational waves (2015, Nobel 2017) confirmed nonlinear solutions of these equations at a distance of 1.3 billion light-years.

Ricci Curvature and Scalar Curvature

Gauss-Bonnet theorem: integral_M K dA = 2pi chi(M). For a torus chi=0, for a sphere chi=2. This is a topological statement: the integral of curvature is independent of the particular metric. Its 4D analogue is the Atiyah-Singer index theorem.

Key ideas

• **Riemann tensor:** R(X,Y)Z=[nabla_X,nabla_Y]Z-nabla_{[X,Y]}Z. For S^2: R_{theta phi theta phi}=sin^2(theta).
• **Ricci tensor:** R_{ij}=R^k_{ikj}. Scalar curvature R=g^{ij}R_{ij}. For S^2: R=2. Bianchi identity: nabla_{[i}R_{jk]lm}=0.
• **Gauss-Bonnet:** integral_M K dA=2pi chi(M). Topological invariant: independent of the specific metric.