I am learning differential geometry and have a few questions on curvature. -- Background:


1. Gauss invented "Gauss curvature" to measure how surface curves.
2. Riemann gives an ingenious generalization of Gauss curvature from surface to higher 
dimensional manifolds using the "Riemannian curvature tensor" (sectional curvature
is exactly the Gauss curvature of the image of the "sectional" tangent 2-dimensional
subspace under the exponential map).
3. In modern textbooks on differential geometry, people usually first define the notion
of connection, and then the Riemannian curvature tensor is expressed in terms of
connection.


My questions are:


1. Are there some other notions of "curvature", besides Gauss curvature and 
Riemannian curvature tensor as its generalization, which people have invented to
measure how space curves?
2. In history, who first introduced the notion of connection to describe
the Riemannian curvature tensor, and why is this idea natural?


Thank you very much!



Are there some other notions of "curvature" which measure how space curves?
in addition to these excellent examples of non-local curvature quantities and their extensions to the non-smooth setting (which I am not sure the OP was anticipating), I might add the 'original' non-local curvature measure: the holonomy.


Also, the OP was not precise about how to interpret the vague term 'space', so it is not at all clear how to answer the question about possible measurements of how 'space curves'. From the context of the question and the given description of the OP's background, I would guess that 'space' is intended to be interpreted as 'Riemannian manifold', in which case the OP is asking some fundamental questions that often aren't explicitly addressed in introductory expositions of Riemannian geometry: First, is the 'Riemann curvature tensor' the whole story in terms of understanding the local geometry of Riemannian manifolds (i.e., in describing how such 'spaces curve')? Second, how did the notion of a 'connection' arise historically, since Riemann didn't use it to define his measure of how 'space curves', and what was the motivation for introducing it? Both these questions frequently occur to beginners in the subject, so it's a bit surprising that they aren't treated a bit more explicitly in introductory books.


For example, consider the following situation: Suppose that, instead of Riemannian geometry on smooth manifolds, we were considering Hermitian geometry on complex manifolds, i.e., our manifolds are complex and we only consider Riemannian metrics that are complex compatible, in the sense that $g(iv) = g(v)$ for all tangent vectors $v$. In this case, the Riemannian curvature tensor of $g$ is *not* the whole story in (complex) dimensions bigger than $1$; in addition to the Riemann curvature tensor, one must consider the exterior derivative of the canonical $2$-form $\omega\_g$ associated to $g$.


It is a nontrivial theorem that, in the case of Riemannian metrics, all of the 'pointwise' differential invariants of a metric $g$' (once these are properly defined) are generated by the 'Riemann curvature tensor' and its 'covariant derivatives'. (Note that this is stronger than what Riemann stated, which is that, if the Riemann curvature tensor vanishes, then the metric is locally Euclidean. It could have been, for example, that, when the Riemann curvature tensor vanishes identically, this overdetermined system of equations forces some other 'independent' invariants to vanish as well, due to integrability conditions of the overdetermined PDE system. As Weyl and Cartan proved, though, this does not happen in the case of Riemannian geometry.) [In the Hermitian case above, it turns out that the Riemann curvature tensor and $d\omega\_g$, together with their 'covariant derivatives' suffice to generate all of the 'pointwise differential invariants of a Hermitian metric $g$ on a complex manifold'. This is also a nontrivial theorem.]


As far as how the notion of connection arose and why it is used so much in Riemannian geometry (and other geometries), that has an interesting history, and I will only sketch it here: In an 1869 paper, Elwin Christoffel showed that one could compute what we now call the Riemann curvature tensor of a metric $g$ as a differential expression in the coefficients of $g$ in two stages. First, one computes certain expressions, the 'Christoffel symbols' $\Gamma^k\_{ij}$, in terms of the coefficients $g\_{ij}$ of $g= g\_{ij} dx^i dx^j$ and their first partial derivatives (with respect to the chosen local coordinate system), and then one computes the Riemann curvature tensor $R^i\_{jkl}$ from the $\Gamma^i\_{jk}$ and *their* first partial derivatives (with respect to the chosen local coordinate system). This gave an efficient method of computing the Riemann curvature tensor and exhibited it explicitly as a second order partial differential expression in the coefficients of $g$ in a somewhat manageable form. 


Then, around 1890, Gregorio Ricci-Curbastro (for whom the Ricci tensor is named), realized that Christoffel's symbols could be used to define a notion of derivative for what we now call tensor fields in the presence of a background Riemannian metric $g$, one that would be independent of any choice of local coordinates, i.e., this derivative would be 'covariant', when one expresses both the tensor field and the metric in local coordinates. He and his former student, Tullio Levi-Civita, used this as the basis of their notion of 'absolute differential calculus'. This led to the idea of tensor fields 'parallel' (i.e., with vanishing absolute derivative) along curves and the notion of 'parallel transport', which gave a way of 'connecting' the tangent spaces along any (piecewise smooth) curve. This original notion of 'connection' (though I am not sure that Levi-Civita actually used this word as a noun) was then vastly generalized and applied to other geometries by Weyl, Schouten, and Cartan. 


The general idea of curvature (and, indeed, holonomy) as a measure of how parallel transport depends on the choice of curve joining two points then became a fundamental notion and, in fact, it turns out that the Riemann curvature tensor of a metric $g$ is precisely the curvature (in this sense of dependence of parallel transport on the path) of the 'connection' introduced by Levi-Civita. [From this point of view, Weyl's theorem that the Riemann curvature tensor and its covariant derivatives give all of the 'pointwise differential invariants' of $g$ is by no means obvious, and it takes some serious work (with the appropriate definitions) to prove it.]


