Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a...

the sectional curvature and e1, ..., en is any orthonormal frame at p. By similar reasoning, the scalar curvature is twice the trace of the curvature operator...

sectional curvature, which is a real number associated to any real 2-plane in the tangent space of X at a point. For example, the sectional curvature...

_{v}R(w,u)+\nabla _{w}R(u,v)=0} Sectional curvature is a further, equivalent but more geometrical, description of the curvature of Riemannian manifolds. It...

nonpositive sectional curvature are joined by a unique geodesic. The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic...

largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved...

analogously, curvature in higher dimensions can be regarded as a kind of tidal force (this is one way of thinking of the sectional curvature). This generalization...

In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely...

component. The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while...

geodesic symmetry at p (less p itself). See (Besse 1978). It has sectional curvature ranging from 1/4 to 1, and is the roundest manifold that is not a...

Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor. The Ricci curvature is determined by the sectional curvatures of a...

plane. All sectional curvatures will have the same sign. If the principal curvatures have different signs: κ1κ2 < 0, then the Gaussian curvature is negative...

centers. For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of...

complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval ( 1 , 4 ] {\displaystyle (1,4]} then...

Riemannian metric of nonnegative sectional curvature may be taken to be closed. Cheeger and Gromoll conjectured that if the curvature is strictly positive somewhere...

corresponding orthonormal eigenvectors (principal directions), then the sectional curvature of M at p is given by K ( X i , X j ) = k i k j {\displaystyle K(X_{i}...

sectional curvature 1 / R 2 {\displaystyle 1/R^{2}} ). However, for n > 1, the Fubini–Study metric does not have constant curvature. Its sectional curvature is...

metric. A Riemannian manifold is said to have constant curvature κ if every sectional curvature equals the number κ. This is equivalent to the condition...

geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number...

simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property...

assumption that the underlying closed Riemannian manifold has nonnegative sectional curvature and parallel Ricci tensor (such as the flat torus or the Fubini–Study...

zero sectional curvature) Jacobi fields are simply those fields linear in t {\displaystyle t} . For Riemannian manifolds of constant negative sectional curvature...

concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is...

metric of constant positive sectional curvature. A 3-manifold with a Riemannian metric of constant positive sectional curvature is covered by the 3-sphere...

numbers, on manifolds which admit Riemannian metrics of nonnegative sectional curvature.[G81a] The principal idea of his work was to combine Karsten Grove...

positive sectional curvature has positive Euler characteristic. A compact, (2d)-dimensional Riemannian manifold with negative sectional curvature has Euler...

the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for positive curvature, geodesics...

whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between...

mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the...

g_{\varepsilon }} is ε {\displaystyle \varepsilon } -flat, i.e. for the sectional curvature of K g ε {\displaystyle K_{g_{\varepsilon }}} we have | K g ϵ | <...