  Here we consider the discrete time dynamics described by a transformation
$T:M \to M$, where $T$ is the shift and $M=\{1,2,...,d\}^\mathbb{N}$. It is
known that the infinite-dimensional manifold $\mathcal{N}$ of H\"older
equilibrium probabilities is an analytical manifold and carries a natural
Riemannian metric. Given a normalized H\"older potential $A$ denote by $\mu_A
\in \mathcal{N}$ the associated equilibrium probability. The set of tangent
vectors $X$ to the manifold $\mathcal{N}$ at the point $\mu_A$ coincides with
the kernel of the Ruelle operator for $A$. The Riemannian norm $|X|=|X|_A$ of
the vector $X$, which is tangent to $\mathcal{N}$ at the point $\mu_A$, is
described via the asymptotic variance, that is, satisfies
  $|X|^2\,\,= \langle X, X \rangle =\lim_{n \to \infty} \frac{1}{n} \int
(\sum_{i=0}^{n-1} X\circ T^i )^2 \,d \mu_A$.
  Consider an orthonormal basis $X_i$, $i \in \mathbb{N}$, for the tangent
space at $\mu_A$. Given two unit tangent vectors $X$ and $Y$ the curvature
$K(X,Y)$ satisfies
  $\,\,\,\,K(X,Y) = \frac{1}{4}[\, \sum_{i=1}^\infty ( \int X \,Y\, X_i \,d
\mu_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,d \mu_A\, \,\int Y^2 X_i \,d \mu_A
\,].$
  When the equilibrium probabilities $\mu_A$ is the set of invariant Markov
probabilities on $\{0,1\}^\mathbb{N}\subset \mathcal{N}$, introducing an
orthonormal basis $\hat{a}_y$, indexed by finite words $y$, we show explicit
expressions for $K(\hat{a}_x,\hat{a}_z)$, which is a finite sum. These values
can be positive or negative depending on $A$ and the words $x$ and $z$. Words
$x,z$ with large length can eventually produce large negative curvature
$K(\hat{a}_x,\hat{a}_z)$. If $x, z$ do not begin with the same letter, then
$K(\hat{a}_x,\hat{a}_z)=0$.
