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 $R\_{ikik}$?



Calculating the Riemann Curvature tensor out of sectional curvature
Not if $n > 2$. The full Riemann tensor has $n^2(n^2-1)/12$ different components. The number of sectional curvatures spanned by two basis vectors is $n(n-1)/2$. The former is always larger than the latter if $n > 2$.


As for estimating the sectional curvature, you might want to study the case $n = 3$ first, because everything reduces to studying a $3$-by-$3$ symmetric matrix. There, your question reduces to the question of whether you can estimate the eigenvalues of such a matrix from knowing its diagonal components. 


