Curvature of a manifold

Question

If we need to find the curvature of a manifold, does that mean we need to find a sectional curvature or a holomorphic sectional curvature in the case of a complex manifold?

It all depends on the context (there is also Ricci and scalar curvatures). — Moishe Kohan, Jun 22 '14 at 00:06

score 2 · Accepted Answer · edited Apr 13 '17 at 12:20

There is no "curvature of a manifold" by itself. You need to specify your settings.

The curvature is a characteristic of a connection, which is an additional structure that can be defined on a manifold. Depending on the way your data are represented, you choose the method, in which you find the curvature. Usually, you have a parametrization or a coordinate patch, in which you obtain the concrete expressions.

You a right in that knowing the sectional curvature suffices to recover the curvature tensor.

With regards to the holomorphic sectional curvature, please refer to the article $[1]$, Proposition 2.2. on p.285 gives the formula expressing the curvature of a nearly Kählerian manifold in terms of the holomorphic sectional curvature.

REFERENCES.

A.Gray, Nearly Kähler manifolds. J. Differential Geometry, 4, 1970, pp.283–309, http://www.ams.org/mathscinet-getitem?mr=0267502

Isn't the Gaussian curvature intrinsic? – user99680 Jul 05 '14 at 04:44 — user99680, Jul 05 '14 at 04:44

Curvature of a manifold

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