Polarization trick

Statement

The polarization trick is a method, or formula, to recover a symmetric bilinear form over the reals from its values on pairs $(X,X)$ where $X$ is a vector.

if $b$ is the bilinear form, the formula is:

$b(X,Y)=1/2(b(X+Y,X+Y)-b(X,X)-b(Y,Y))$

Note that if the space is endowed with a Euclidean inner product, it suffices for us to know the value on pairs $(X,X)$ where $X$ is a unit vector, because every vector is a scalar multiple of a unit vector.

The polarization trick is used, for instance, to show that:

• The Ricci curvature determines the Ricci curvature tensor
• The sectional curvature determines the Riemann curvature tensor