Let $(M, g)$ be a Riemannian manifold. Define the curvature tensor convention as follows.


$$ R(X, Y) Z = \nabla\_X \nabla\_Y Z - \nabla\_Y \nabla\_X Z - \nabla\_{[X,Y]} Z$$


$$ R(X,Y,Z,W) = g(R(X,Y)Z, W)$$


It is well-known that the curvature tensor $R$ is explicitly expressed by the sectional curvatures. This can be found in Jost's
*Riemannian Geometry and Geometric Analysis*.



> 
> **Lemma.** With $K(X,Y) = R(X, Y, Y, X)$,
> \begin{align} R(X,Y,Z,W) = & + K(X+W, Y+Z) - K(X+W, Y) - K(X+W, Z) \\ &- K(X, Y+Z) - K(W,Y+Z) + K(X,Z) + K(W,Y) \\ & - K(Y+W, X+Z) + K(Y+W, X) + K(Y+W,Z) \\ & + K(Y,X+Z) + K(W,X+Z) - K(Y,Z) - K(W,X).\end{align}
> 
> 
> 


My question is whether there is a similar explicit formula for the Riemannian curvature tensor and *the holomorphic sectional curvature* when we assume that the manifold is Kähler.


I think [this MO question](https://mathoverflow.net/questions/79383/calculating-a-curvature-tensor-by-polarization) is related to my question. From this polarization formula, at least, we know that the curvature tensor is determined algebraically by holomorphic sectional curvatures. However, I am curious about whether there is a (relatively) simpler formula describing the algebraic relation between the curvature tensor and holomorphic sectional curvatures.


Thanks!



About an explicit formula of the curvature tensor by holomorphic sectional curvatures
Following the suggestion by @YangMills, I used Mathematica to combine the two formulas to get the full expression. As expected, the resulting formula is quite complicated. However, there is one neat property, which is that the formula for $R(X,Y,Z,W)$ does not contain any terms of the form $H(X), H(Y), H(Z)$ or $H(W)$.


This is intuitive in hindsight, because $R(X,Y,Z,W)$ must be anti-symmetric with respect to the $(X,Y)$-pair as well as the $(Z,W)$-pair. A non-zero term of the form $H(X)$ would break this anti-symmetry. However, I didn't realize until after computing the full expression, so thought it was worth providing an answer.


The full formula for $R(X,Y,Z,W)$ is the following. Here, $X,Y,Z,W$ are all real vectors and $H(X)=R(X,JX,JX,X)$ is the holomorphic sectional curvature.


$$R(X,Y,Z,W) = \frac{1}{32}\left( H[W-X]+5 H[W+X]-3 H[W-J X]-3 H[W+J X]-H[W-Y]+H[W+X-Y]-5 H[W+Y]-H[W-X+Y]+3 H[W-J X+Y]+3 H[W+J X+Y]+3 H[W-J Y]-3 H[W+X-J Y]+3 H[W+J Y]-3 H[W+X+J Y]-H[W-X-Z]-H[X-Z]+H[W+X-Z]+H[W-Y-Z]+H[X-Y-Z]-H[W+X-Y-Z]+H[Y-Z]-H[W+Y-Z]-H[-X+Y-Z]+H[W-X+Y-Z]-5 H[X+Z]+5 H[Y+Z]+3 \left( H[X-J Z]-H[W+X-J Z]-H[Y-J Z]+H[W+Y-J Z]+H[X+J Z]-H[W+X+J Z]-H[Y+J Z]+H[W+Y+J Z]+H[W-J (X+Z)]+H[Y-J (X+Z)]-H[W+Y-J (X+Z)]+H[W+J (X+Z)]+H[Y+J (X+Z)]-H[W+Y+J (X+Z)]-H[W-J (Y+Z)]-H[X-J (Y+Z)]+H[W+X-J (Y+Z)]-H[W+J (Y+Z)]-H[X+J (Y+Z)]+H[W+X+J (Y+Z)]) \right) \right) $$


