A pseudo-Riemannian manifold is a smooth manifold with a smooth metric tensor that is a nondegenerate bilinear form on each tangent space. Positivity is not required. A Riemannian manifold is the positive-definite case, while a Lorentzian manifold has exactly one temporal direction in its metric signature.
Geometric splitting theorems give hypotheses under which a manifold decomposes as a metric product. The Cheeger-Gromoll splitting theorem gives a Euclidean line factor for a complete connected Riemannian manifold with nonnegative Ricci curvature containing a globally minimizing geodesic line. Other splitting theorems use different curvature or causal hypotheses.
A pseudo-Riemannian metric is a smooth nondegenerate symmetric bilinear form on every tangent space, with locally constant signature. Its positive-definite special case is a Riemannian metric. Unlike a positive-definite metric, an indefinite metric admits nonzero null vectors. In either signature its Levi-Civita connection is uniquely determined by torsion-freeness and metric compatibility.
A Ricci-flat manifold has vanishing Ricci tensor. This is weaker than vanishing Riemann curvature tensor: the latter describes complete local flatness. A Kerr black hole supplies a Lorentzian manifold that is Ricci flat in its vacuum region but has nonzero Riemann curvature tensor.
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A pseudo-Riemannian manifold is a generalization of a Riemannian manifold that allows for the metric tensor to have signature that is not positive definite. While in a Riemannian manifold the metric tensor \( g \) is positive definite, which means that for any nonzero tangent vector \( v \), the inner product \( g(v, v) > 0 \), a pseudo-Riemannian manifold has a metric tensor that can have both positive and negative eigenvalues.