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In many papers of semi-Riemannian geometry, when they talk about curvature (constant or bounded) they don't precice which kind of curvature they talk about. I know the definition of:

-Sectional curvature

-Gaussian curvature

-Mean curvature

-Principal curvature

-Ricci curvature

(and there is a genetral one, called the curvature in the sense of A.D. Alexandrov using comparison triangles).

But, how to know from the context, which curvature the author talk about? (i think may be with smooth metrics, it's always sectional curvature? ).

When these curvatures coincid (are equal) ?

when we say for example "a manifold with constant curvature" is this always "sectional curvature ? thanks for discussing.

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  • $\begingroup$ Also essential is the ([semi-]Riemannian) curvature tensor $R \in \Gamma(\bigwedge^2 T^*M \otimes L(TM, TM))$; the sectional curvature and Ricci curvature are usually defined in terms of $R$. $\endgroup$ Jul 21, 2020 at 22:50
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    $\begingroup$ Usually yes, they mean the sectional curvature. If it is some other curvature, they should say explictly what it is. Constant sectional curvature is a strong condition because it implies that (up to a sign that depends on conventions, but I don't want to get political here) $$R(X,Y)Z = K(\langle Y,Z\rangle X - \langle X,Z\rangle Y),$$and this implies ${\rm Ric} = (n-1)K\langle\cdot,\cdot\rangle$ and ${\rm s} = n(n-1)K$. $\endgroup$
    – Ivo Terek
    Jul 21, 2020 at 23:01
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    $\begingroup$ Mean curvature is only meaningful if your manifold is embedded somewhere else, and you want to discuss minimal submanifolds, marginally trapped submanifolds or whatnot. Principal curvatures also depend on an embedding, and if the codimension of the embedding is greater than 1, it is less interesting because you have lots of choices of normal directions. $\endgroup$
    – Ivo Terek
    Jul 21, 2020 at 23:03
  • $\begingroup$ @IvoTerek thank you for the answer its more clear for me now! What about the relation between the Gaussian curvature and the sectional curvature ?? are they equal in somme cases ? for example on smooth surfaces ? (may be i am saying a stupidity ...) or for example, a paper says: the gaussian curvature of a convex surface in a Lorentzian manifold is always positive!! what about the sectional curvature in this case ? is it also positive ? maybe there is a relation between the two notions ? $\endgroup$ Jul 22, 2020 at 13:31
  • $\begingroup$ The Gaussian curvature is just a name for the sectional curvature in dimension 2. If the surface is embedded in $\Bbb R^3$, it coincides with the product of principal curvatures (this is the Theorema Egregium). I absolutely do not mean to discourage you but I do think you would greatly benefit from taking a Riemannian geometry course to avoid getting distracted with terminology issues that are unimportant in the big scheme of things. $\endgroup$
    – Ivo Terek
    Jul 22, 2020 at 15:17

1 Answer 1

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As the name suggests, they would be talking about the Riemannian curvature (cf. Riemann tensor).

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