Let $M$ be a $C^{\infty}$ manifold. We can define a contrast function $\rho:M\times M\to \mathbb{R}$ satisfies $\rho(x,y)\geq0$ for all $x,y\in M$ and equality holds iff $x=y$.
In Eguchi's 1991 paper "Geometry of minimum contrast",he defines a Riemannian metric on M by $G(X,Y)=-\rho(X|Y)$ where $X,Y$ are vector fields on M. The notation $\rho(X|Y):=X_{x}Y_{y}\rho(x,y)$ where $ X_{x}=X^{i}_{x}\bigg(\frac{\partial}{\partial x_{i}}\bigg)_{x},Y_{y}=Y^{i}_{y}\bigg(\frac{\partial}{\partial x_{i}}\bigg)_{y}.$
I am trying to prove G as a Riemannian metric for that I need to prove
$\bullet$ G is bilinear. Which can be prove from the definition.
$\bullet$G(X,X)>0 for all $X\neq 0 .$
$\bullet$G(X,Y)=G(Y,X).
To prove positive-definiteness and symmetry properties we need the following two results
$\bullet \rho(Y| \cdot)=0$ for any vector field $Y$.
$\bullet$ $\rho(XY|\cdot)=-\rho(X|Y).$
I am unable to get these two and following is what I am trying:
Since $\rho (x,y)$ has minimum value $0$ at $x=y$ therefore we have
$\bigg(\frac{\partial}{\partial x_{i}}\bigg)_{x}\rho(x,y)\bigg|_{y=x}=0$ and $\bigg(\frac{\partial}{\partial x_{i}}\bigg)_{y}\rho(x,y)\bigg|_{y=x}=0$
Let $X,Y$ be two any vector fields over M.
Then $\rho(Y|X)=Y_{x}X_{y}\rho(x,y)\bigg|_{y=x}=0$.
That is $\rho(Y| \cdot)=0$ for any vector field $Y$.
Applying $X$ to $\rho(Y| \cdot)=0$ ,we have
$X\bigg(\rho(Y| \cdot)\bigg)=0$
$\implies X(Y_{x}\rho(x,y)\bigg|_{y=x})=0.$
The followings are from the paper
How should I go to get the results?Is my way correct or am I mising something? I will be thankful if anyone can give me some hints on it.