TITLE: Calculating the Riemann Curvature tensor out of sectional curvature
QUESTION [8 upvotes]: Hi everybody,
my question is the following: Let $(M^n,g)$ be a Riemannian manifold and $e_1,\ldots,e_n$ be an orthonomal frame in a point. Assume, that we now the sectional curvatures of all planes, spanned by these vectors, i.e. we know the components $R_{ijij}$ of the curvature tensor.
Is it then possible to calculate all other components $R_{ijkl}$?
The problem is, that if I want, e.g. to calculate $R_{ijik}$ by the polarization identity, I need to know the sectional curvature spanned by $e_1,e_j+e_k$ which I assume to be unknown.
Is it then possible to calculate, or at least to estimate the sectional curvature from above by $R_{ijij}$, $R_{jkjk}$ and $R_{ikik}$?

REPLY [7 votes]: There is an explicit, but complicated, formula for going from the sectional curvature back to the curvature tensor in equation (1.10) on page 16 of Cheeger and Ebin's book.