In a paper by Yann Ollivier:

Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from $x$ to $y$ along $v$. If, instead of a Riemannian manifold, we were working in ordinary Euclidean space, the endpoints $x_0$ and $y_0$ of $w_x$ and $w_y$ would constitute a rectangle with $x$ and $y$. But in a manifold, generally these four points do not constitute a rectangle any more.

Indeed, because of curvature, the two geodesics starting along $w_x$ and $w_y$ may diverge from or converge towards each other. Thus, on a sphere (positive curvature), two meridians starting at two points on the equator have parallel initial velocities, yet they converge at the North (and South) pole. Since the initial velocities $w_x$ and $w_y$ are parallel to each other, this effect is at second order in the distance along the geodesics (Fig.).

Thus, let us consider the points lying at distance $\varepsilon $ from $x$ and $y$ on the geodesics starting along $w_x$ and $w_y$, respectively. In a Euclidean setting, the distance between those two points would be $|v|$, the same as the distance between $x$ and $y$. The discrepancy from this Euclidean case is used as a definition of a curvature.

**Definition**(Sectional curvature). Let $(X, d)$ be a Riemannian manifold.
Let $v$ and $w_x$ be two unit-length tangent vectors at some point $x \in X$. Let $\varepsilon, \delta > 0$.
Let $y$ be the endpoint of $v$ and let $w_y$ be obtained by parallel transport of $w_x$ from $x$ to $y$. Then
$$ d(exp_{x} \varepsilon w_x, exp_{y} \varepsilon w_y) = \delta (1 −\frac{\varepsilon^2}{2} K(v,w)+ O(\varepsilon^3 +\delta \varepsilon^2))
$$

when $\delta , \varepsilon \to 0$. This defines a quantity $K(v,w)$, which is the *sectional curvature*
at $x$ in the directions $(v,w)$.

**Question1**
How can I derive the formula $ d(exp_{x} \varepsilon w_x, exp_{y} \varepsilon w_y) = \delta (1 −\frac{\varepsilon^2}{2} K(v,w)+ O(\varepsilon^3 +\delta \varepsilon^2))
$ from figure?

**Question2**
How this definition of sectional curvature can be derived from its usual definition ($K(v,w)=\frac{\langle R(v,w)w, v\rangle}{\langle v,v\rangle \langle w,w \rangle - \langle v,w \rangle ^2}$)

Thanks in advance for your time.