Sectional curvature
Sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature < ...
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Talk:Sectional curvature
Talk:Sectional curvature Hi. Sectional Curvature. I can see that a number of ... technical layman. My question. The explanation of sectional curvature that appears currently is no ...
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Ricci curvature
Ricci curvature It has been suggested that Trace-free ... section. (Discuss) In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is ... obtained as a trace of the full curvature tensor. It can be thought of as ... u where R is the Riemann curvature tensor. In local coordinates, one can write ... is the only independent trace of the curvature tensor; all others vanish or are ...
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Constant curvature
Constant curvature In mathematics, constant curvature in differential geometry is a concept most ... applied to surfaces. For those the scalar curvature is a single number determining the local ... at all points. The circle has constant curvature, also, in a natural (but different) sense. The standard surface geometries of constant curvature are elliptic geometry (or spherical geometry) ...
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Gaussian curvature
Gaussian curvature In mathematics, the Gaussian curvature of a point on a surface is ... of the given point. Symbolically, the Gaussian curvature K is defined as $K\; =\; k\_1\; ...\; math>\backslash mathbb\{R\}^3$, the Gaussian curvature is also given by $K(\backslash mathbf\; ...\; S\; is\; the\; shape\; operator.\; See\; also$sectional curvature Mean curvature
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Riemann curvature tensor
Riemann curvature tensor In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any ... affine connection, torsionless or with torsion. The curvature tensor is given in terms of a ... each argument. NB. Some authors define the curvature tensor with the opposite sign. If < ...
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Curvature of Riemannian manifolds
Curvature of Riemannian manifolds In mathematics, specifically differential ... more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions. The curvature of a Pseudo-Riemannian manifold can be ... only slight modifications. Ways to express the curvature of a Riemannian manifold The curvature ...
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Talk:Curvature of Riemannian manifolds
Talk:Curvature of Riemannian manifolds Sectional curvature part, first paragraph: there are some missing ... you catch the most general aspect of curvature in the sense of Nomizu, Kulkarni and ... a decomposition of the space of all curvature structures in terms of Lie and ...
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Riemannian geometry
... Metric tensor Riemannian manifold Levi-Civita connection Curvature Curvature tensor. The following articles might be also ... Bonnet Theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold ... behavior of the space (usually formulated using curvature assumption) to derive some information about the ... of points at "sufficiently large" distances. Pinched sectional curvature 1/4-pinched Sphere Theorem. ...
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Talk:Soul theorem
... statement of Gromoll and Cheeger that pointwise sectional curvature anywhere in M is enough to send ... part of the proof refers to local curvature restrictions? From what I can tell, we ... A, for example, explicitly states a nonnegative curvature assumption, not positive curvature. I added the corollary because it ...
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