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A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field).
The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let A {\displaystyle A} and B {\displaystyle B} be symmetric covariant 2-tensors.
Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold M, the length of a smooth curve between two points p and q can be defined by integration, and the distance between p and q can be defined as the infimum of the lengths of all such curves; this makes M a metric space.
Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
Suppose that (,) is an -dimensional Riemannian or pseudo-Riemannian manifold, equipped with its Levi-Civita connection.The Riemann curvature of is a map which takes smooth vector fields , , and , and returns the vector field (,):= [,] on vector fields,,.
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ()Riemannian metric and is torsion-free.
A conformal manifold is a Riemannian manifold (or pseudo-Riemannian manifold) equipped with an equivalence class of metric tensors, in which two metrics g and h are equivalent if and only if =, where λ is a real-valued smooth function defined on the manifold and is called the conformal factor.