Levi-Civita connection

The unique connection on a Riemannian manifold that preserves its metric and has zero torsion. It determines parallel transport and geodesics from the metric alone.

Metric compatibility: A property of a connection that makes covariant differentiation preserve the metric tensor. One of the two conditions that uniquely characterize the Levi-Civita connection.
Riemannian manifold: A smooth manifold equipped with a smoothly varying positive-definite inner product on each tangent space. The Levi-Civita connection is defined for this metric geometry.
Weitzenböck connection: A flat connection with torsion, often defined using a global frame on a parallelizable manifold. It exchanges the Levi-Civita connection's zero torsion for nonzero torsion and zero curvature.
Riemannian geometry: The study of smooth manifolds equipped with Riemannian metrics and their geometric properties. The Levi-Civita connection is the standard differential structure underlying this geometry.
Tullio Levi-Civita: An Italian mathematician known for foundational work in tensor calculus and differential geometry. His name became attached to the canonical metric-compatible, torsion-free connection.
Torsion tensor: A tensor measuring the antisymmetric part of a connection's covariant derivative of vector fields. Its vanishing supplies the connection's second defining condition.
Affine connection: A rule for differentiating vector fields along vector fields on a smooth manifold. The Levi-Civita connection is a uniquely specified affine connection.
Levi-Civita connection with torsion: A metric-compatible connection allowed to have nonzero torsion; despite the name, it is not Levi-Civita in the strict sense. It retains metric compatibility while relaxing the torsion-free condition.
Riemannian distance: The distance between points defined as the infimum of lengths of curves joining them in a Riemannian manifold. Geodesics of the connection locally realize this metric distance.
Gregorio Ricci-Curbastro: An Italian mathematician who developed absolute differential calculus, later known as tensor calculus. His tensor calculus supplied tools for expressing metric connections invariantly.

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