Retrieving "Sectional Curvature" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Ricci Tensor

   Linked via "sectional curvature"

   The Ricci Tensor in Non-Standard Geometries
   While most commonly associated with Lorentzian manifolds (spacetime), the Ricci tensor is a well-defined concept in Riemannian manifolds (space). In positively curved spaces, such as the 3-sphere ($S^3$) embedded in Euclidean space, the Ricci tensor is directly proportional to the metric tensor ($R{\mu\nu} = \lambda g{\mu\nu}$), a property known as being Einsteinian. This propor…

2. Saddle Point

   Linked via "sectional curvature"

   Geometric Interpretation (Manifolds)
   In differential geometry, a saddle point corresponds to a point on a manifold where the Gaussian curvature is negative. This is quantified by the sectional curvature, which must take opposing signs along orthogonal 2-dimensional planes passing through the point. If the function is visualized, the surface locally resembles a horse saddle or a Pringle chip (though the latter is a less formal term).
   The tend…