Riemann Curvature Not Along Geodesics? In calculus you're sometimes told that the second derivative $y''$ at a critical point determines the curvature of a curve. Studying further we find this is true because the formula for curvature $$k = \frac{y''}{(1 + (y')^2)^{3/2}},$$
when $y' = 0$, is just $y''$. In differential geometry, you can derive the Riemann curvature tensor 
\begin{align}
&R^a_{bcd} = - (\Gamma^a_{bcd} - \Gamma^a_{bdc} + \Gamma_{bc}^e \Gamma_{de}^a - \Gamma^e_{bd} \Gamma^a_{ce}) \\
&[\text{Note the nice} \ -(bcd - bdc + bc^e d_e - bd^e c_e) \ \text{pattern!}]
\end{align}
by deriving the equation of geodesic deviation (Jacobi fields) on $v^a = \frac{\partial x^a(t,s)}{\partial s}$ on geodesics $x^a(t)$ parametrized by $s$, $x_s(t) = x(t,s)$, $u^a = x^a_t$:
$$\frac{D^2 v^a}{Dt^2} = - R^a_{bcd} u^b u^c v^d,$$
which shows the second covariant derivative along a curve determines the curvature along geodesics, just as the second derivative of $y = f(x)$ determines the curvature at a critical point.
Is there an analogue of $k = \frac{y''}{(1 + (y')^2)^{3/2}}$ for the Riemann curvature, that reduces to $R^a_{bcd}$ along geodesics?
Sectional curvature kind of looks like it but this formula reduces to it for orthogonal vectors, no mention of geodesics.
 A: The two notions of curvature that you'd like to compare are conceptually different.
The Riemann tensor describes the intrinsic curvature of a manifold, measuring (roughly speaking) the failure of ordinary flat Euclidean geometry: how geodesics that start parallel diverge or converge, how for a small circle the ratio of circumference to radius deviates from $2\pi$, and so forth. "Intrinsic" means that these notions all make sense within the manifold itself.
The curvature of the graph of a function that you quote as
$$
k = \frac{y''}{(1 + (y')^2)^{3/2}}
$$
is an extrinsic notion of curvature, which tells you how one manifold (your curve $y=y(x)$ in this case, the "submanifold") is embedded in another (the flat two-dimensional plane here). This notion of curvature tells you, for example, how quickly a geodesic in the ambient space diverges from a geodesic within the submanifold. It's extrinsic because it makes sense only if you refer to the space outside the curve, and how the curve lies in that higher-dimensional space.
To mark the difference, intrinsically a plane or curve or cylinder are all identical (you can make them from a flat sheet of paper), but they differ in extrinsic curvature, which refers to how they're embedded in the ambient three-dimensional space.
The object in differential geometry that generalises $k$ is a tensor sometimes called the second fundamental form. It tells you how, if you start with a vector tangent to a submanifold and parallel transport it by another tangent vector, it moves off the submanifold. Geodesics have vanishing second fundamental form, because they parallel transport their own tangent vector: they're the "straight lines" on the manifold, with no extrinsic curvature.
