In your setting, "constant curvature" means constant sectional curvature. The usual definition is "every tangent $2$-plane has the same curvature"; the formula you give for the Riemann curvature tensor is then a theorem. In differential geometry, a metric with constant Ricci curvature is usually called an Einstein metric.

2016年9月30日 · In particular, the case you propose, $\tau(s) = e^s$ might be solvable explicitly, and hopefully you can actually find the curve with this torsion and constant curvature, but it is also possible that the differential equations of the Frenet-Serret frame in this case are not exactly solvable (this is a system of linear odes with varying ...

Einstein 3-manifold has constant sectional curvature Hot Network Questions Some instruments (e.g. banjo and electric bass) can be fretted or fretless, other instruments are always fretted or always fretless.

Added: If it were possible to wind a curve around a cylinder, and to bend the cylinder back to itself - while keeping the curvature of the helix constant by appropriate adjustments -, thus forming an eventually arbitrarily knotted torus with a closed helix on it, one could have every knot as a curve of constant curvature.

2015年6月12日 · Are there surfaces that have (except for cusps or borders) a constant positive gaussian curvature but that do not have not a constant mean curvature? 1 Evans Chap 8.12 - Constant Mean Curvature Surface and Variations

Constant curvature. Ask Question Asked 7 years, 11 months ago. Modified 7 years, 11 months ago. Viewed 1k ...

2016年4月6日 · Theorem (Killing-Hopf): a simply connected complete surface with constant scalar curvature is isometric to either the 2-sphere with its round metric; the plane with its Euclidean metric; or the disc with its hyperbolic metric. Thus $\tilde \Sigma$ is conformally isometric to one of the three examples; hence is biholomorphic to it.

2020年3月8日 · And thus every constant curvature space can be written as a locally symmetric space as $$ \Gamma \backslash G/ H $$ where $ \Gamma $ is a discrete subgroup of isometries acting freely on $ X=G/H $ (here $ G $ is the isometry group of $ X $, and $ X $ is the simply connected symmetric space of the appropriate curvature).

2019年3月13日 · $\begingroup$ Even in Riemannian geometry, there are different notions of "constant curvature", at least you have constant sectional curvature (which characterizes space-forms), covariantly constant curvature (which characterizes locally symmetric spaces) and constant scalar curvature (which probably is to weak to count as a condition of constant …

2021年1月19日 · I always had the impression that this is also an equivalent condition, i.e $$ R_{ijkl;m}=0\Longleftrightarrow (M,g)\text{ is of constant curvature}, $$ however I do not know any proof and I skimmed through quite a few books on Riemannian geometry both modern and old-school and I did not see this statement or a proof of it.