Sectional Curvature

For any smooth vector field v.

Sectional curvature. It is the gauss curvature of the section at p. Given a riemannian manifold and two linearly independent tangent vectors at the same point u and v we. A riemannian manifold viz a differential manifold equipped with a riemannian metric.

And of course it wasn t long before christoffel wrote down formulas for sectional curvature which don t depend on any embedding. Sectional curvature is a further equivalent but more geometrical description of the curvature of riemannian manifolds. The sectional curvature determines the curvature tensor completely.

We now pull out of our hat the tensor operatorname rm x y z w langle x z rangle langle y w rangle langle x w rangle langle y z rangle or recall where the idea of constant sectional. Sectional sofas are made with sleek linear lines and have a rectangular or square form. The proposed definition of sectional curvature above isn t actually viable but it is historically part of how the theory got off the ground.

Here section is a locally defined piece of surface which has the plane as a tangent plane at p obtained from geodesics which start at p in. If you want to create more interest juxtapose a round. Let be a tangent plane to at a point then the sectional curvature of at is defined as follows.

Sectional curvature 403 vector field v x m is a vector field along α iff v x y t α x y m forall x y u. Let r ij denote the ricci tensor and let q be the quadratic form on t p m n given by r ij at p in m n. To balance these shapes a coffee table can either mimic the planes of the sectional or the table can add contrast.

In riemannian geometry the sectional curvature is one of the ways to describe the curvature of riemannian manifolds the sectional curvature k σ p depends on a two dimensional plane σ p in the tangent space at p it is the gaussian curvature of the surface which has the plane σ p as a tangent plane at p obtained from geodesics which start at p in the directions of σ p in other words the. It is a function which depends on a section i e. Thus the sectional curvature determines the curvature tensor because a quadratic form determines a unique bilinear form by polarization.