Differential geometry MOC

Semi-Riemannian manifold

A semi-Riemannian -manifold is a -manifold equipped with a special kind of tensor field called a metric tensor. diff This means must satisfy

1. symmetry, i.e. ;
2. non-degeneracy, i.e. iff ;

Note that by non-degeneracy we can define the inverse metric tensor so that . See Raising and lowering of indices.

Further terminology

Locally the metric tensor can be brought into the form of a diagonal matrix with entries . For a connected manifold, the numbers of positive and negative entries is an invariant , called the signature.

• gives a Riemannian manifold
• gives a Lorentzian manifold

See also

• Christoffel symbols

---

tidy | en | sembr