Let and be the curvature tensors of conjugate connections and on , respectively. Then
Proof First consider the left hand side: Since and are conjugate connections, we have for all . Therefore, $$\begin{aligned}
&\langle R(X,Y)Z,W\rangle \ =& \langle \nabla_{X}\nabla_{Y}Z,W\rangle - \langle \nabla_{Y}\nabla_{X}Z,W\rangle - \langle \nabla_{[X,Y]}Z,W\rangle \
=& (X\langle\nabla_{Y} Z,W\rangle - \langle\nabla_{Y}Z,\tilde{\nabla}{X}W\rangle) - (Y\langle\nabla{X} Z,W\rangle - \langle\nabla_{X}Z,\tilde{\nabla}{Y}W\rangle) \
&-( [X,Y]\langle Z,W\rangle -\langle Z, \tilde{\nabla}{[X,Y]}W\rangle) \
=& XY\langle Z,W\rangle -X \langle Z, \tilde{\nabla}{Y}W\rangle - Y\langle Z , \tilde{\nabla}{X}W\rangle + \langle Z, \tilde{\nabla}{Y}\tilde{\nabla}{X}W\rangle - YX\langle Z,W\rangle + \
&Y\langle Z, \tilde{\nabla}{X}W\rangle + X\langle Z, \tilde{\nabla}{Y}W\rangle - \langle Z, \tilde{\nabla}{X}\tilde{\nabla}{Y}W\rangle - [X,Y]\langle Z,W\rangle + \langle Z, \tilde{\nabla}{[X,Y]}W\rangle \
=& \langle Z, \tilde{\nabla}{Y}\tilde{\nabla}{X}W\rangle -\langle Z, \tilde{\nabla}{X}\tilde{\nabla}{Y}W\rangle +\langle Z, \tilde{\nabla}{[X,Y]}W\rangle\
=& - \langle Z, \tilde{R}(X,Y)W\rangle
\end{aligned}$$$\square$
Fundamental Theorem of Information Geometry
A torsion-free affine connection has constant curvature if and only if its conjugate torsion-free connection has the same constant curvature .
Proof
A statistical manifold is said to be -flat if the induced -connections are flat.