Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. 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. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. 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. For a more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as the differential geometry of surfaces. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications. The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiation) ∇ {displaystyle abla } and Lie bracket [ ⋅ , ⋅ ] {displaystyle } by the following formula: Here R ( u , v ) {displaystyle R(u,v)} is a linear transformation of the tangent space of the manifold; it is linear in each argument.If u = ∂ / ∂ x i {displaystyle u=partial /partial x_{i}} and v = ∂ / ∂ x j {displaystyle v=partial /partial x_{j}} are coordinate vector fields then [ u , v ] = 0 {displaystyle =0} and therefore the formula simplifies to i.e. the curvature tensor measures noncommutativity of the covariant derivative. The linear transformation w ↦ R ( u , v ) w {displaystyle wmapsto R(u,v)w} is also called the curvature transformation or endomorphism. NB. There are a few books where the curvature tensor is defined with opposite sign.