Top 10 similar words or synonyms for scalar_curvature

ricci_curvature    0.879283

sectional_curvature    0.875758

laplace_operator    0.862710

dirac_delta_function    0.862146

riemann_curvature_tensor    0.859268

laplace_beltrami_operator    0.857543

ricci_tensor    0.857492

metric_tensor    0.853358

laplacian    0.852242

covariant_derivative    0.850282

Top 30 analogous words or synonyms for scalar_curvature

Article Example
Scalar curvature In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.
Scalar curvature For a closed Riemannian 2-manifold "M", the scalar curvature has a clear relation to the topology of "M", expressed by the Gauss–Bonnet theorem: the total scalar curvature of "M" is equal to 4 times the Euler characteristic of "M". For example, the only closed surfaces with metrics of positive scalar curvature are those with positive Euler characteristic: the sphere "S" and RP. Also, those two surfaces have no metrics with scalar curvature ≤ 0.
Scalar curvature Conversely, Stolz showed that every simply connected spin manifold of dimension at least 5 with α-invariant zero has a metric with positive scalar curvature.
Scalar curvature The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first raise an index to obtain a (1,1)-valent tensor in order to take the trace. In terms of local coordinates one can write
Scalar curvature Thus, the second derivative of this ratio, evaluated at radius "ε" = 0, is exactly minus the scalar curvature divided by 3("n" + 2).