Although spectacular in terms of achievement, development of humanity's understanding of curvature has been laborious and slow. Meanwhile, the tensor framework will make most of the concepts and equations appear deceptively simple as many of the most profound and hard-fought facts will fall before us like dominos. For example, Gauss's famous Theorema Egregium -- the Remarkable Theorem -- will come across as almost entirely self-evident. His celebrated equations of the surface will be derived in an exercise in the next Chapter. Meanwhile, Euler's remarkable discovery that the surface swept out by tangents to a three-dimensional curve is developable will, too, be found in an exercise. In fact, most of the facts that might occur to us naturally, some found in exercises, would have constituted results worthy of an admirable nineteenth-century doctoral thesis.

Such retrospective ease is not unique to Tensor Calculus but is a general phenomenon of human forward movement. In music, it is said, the dissonance of today is the consonance of tomorrow. This law of evolution of human thought should in no way diminish our admiration for the remarkable achievements of the great minds of the past. As we celebrate our superior tools of the trade, we should contemplate what has been lost because of them.

Let us begin by having an informal discussion of curvature and, in particular, its interplay with coordinates systems. The crucial observations made in this Section, although stated as facts, are meant to appeal to your geometric intuition and to guide our future analytical investigations.

In Introduction to Tensor Calculus, we devoted much of our attention to Euclidean spaces characterized by straightness. A Euclidean space can accommodate a straight line in any direction and can therefore be referred to an affine -- and, in particular, Cartesian -- coordinate system. Nevertheless, in almost all of our theoretical explorations we assumed a more general curvilinear coordinate system. Our primary motivation for this choice was to eliminate the possible artifacts of a special coordinate system. Thus, we did not assume that the chosen coordinate system had any special features, such as straight regularly-spaced coordinate lines. When we did, on rare occasions, resort to affine coordinates, it was to specifically call attention to the Euclidean nature of our space. Our diligent self-limitation paid off as it led us to develop a more general framework of great robustness.

However, in order to arrive at such a framework, we had to first contend the curvature effects associated with curvilinear coordinates. The most obvious and striking complication lies in the fact that the covariant basis $\mathbf{Z}_{i}$, which is regular (i.e. constant) in affine coordinates, varies from one point to the next in curvilinear coordinates. As a result, vectors at different points cannot be compared, added, or otherwise related in any way by analyzing their components in the familiar Cartesian fashion. Furthermore, the partial derivative applied to the components of a vector field does not fully capture its rate of change since it ignores the variability in the accompanying basis. This loss of geometric information manifested itself in the lack of the tensor property of the partial derivative of a tensor.

Outside of our geometric study of curves, this was our first encounter with the effects of curvature. However, this curvature is of the spurious nature: since a Euclidean space is fundamentally straight, these effects are artifacts of our analysis and, as a result, can be completely mitigated. This is accomplished by introducing the covariant derivative $\nabla_{i}$. The covariant derivative restores the tensor property, possesses the metrinilic property with respect to the basis, i.e. which is akin to the affine property and preserves the product rule.

In summary, the tensor framework handles this lowest "degree" of curvature rather easily while offering a great deal of insight.

The next "degree" of curvature is represented by surfaces that can be formed by smoothly curving a flat sheet of paper, i.e. a region of a plane, without any distortion. For a transformation to not introduce any distortion it must preserve the distance between any two points as measured within the surface. We must note, however, that the concept of distance as measured within the surface is arguably unclear and needs to be precisely defined. Nevertheless, such transformations are known as isometric and surfaces that can be isometrically transformed into a region of a plane are known as developable. Thus, we can continue to think of a developable surface as a Euclidean plane from the viewpoint of their internal geometry but not from the overall three-dimensional point of view.

Developable surfaces were first investigated by Euler in his 1772 work De solidis quorum superficiem in planum explicare licet or On shapes that can be unwrapped onto a plane. The classical examples of developable surfaces are the cylinder and the cone. Note that these two surfaces need to be cut in order to be transformed into a region of a plane.

Another general type of a developable surface is known as the tangent developable of a curve. It is the surface swept out by the tangent lines to a three-dimensional curve. In De solidis quorum superficiem in planum explicare licet, Euler not only demonstrated that the tangent developable is, indeed, a developable surface, but that every developable surface that is not a cylinder or a cone is a tangent developable for some curve.

On the one hand, developable surfaces are undeniably curved. On the other hand, they allow regular coordinate systems which can be accomplished easily by referring the flat sheet of paper to a regular coordinate system prior to curving it into the final shape, as illustrated in the following figure. Of course, we must clarify the sense in which a coordinate system on the curve surface is regular. Indeed, in the above figure, all of the coordinate lines are curved and the coordinate nodes are not arranged in any kind of regular pattern in the three-dimensional space. Yet, it is intuitively clear that, with respect to the surface, the coordinate lines are indeed regular. Thus, there is a clear dichotomy between how the image appears in the overall three-dimensional space and how it is perceived within the surface. We will use the term intrinsic to describe the geometric action that takes place strictly in the surface without a reference to ambient objects. For example, we will describe developable surfaces as curved in the three-dimensional sense and as flat in the intrinsic sense.

A surface swept by straight lines (which are not necessarily the tangents of a given curve) is known as a ruled surface. We will show that every developable surface is ruled. This becomes intuitively clear when inspecting a curved sheet of paper and noting that, at every point, the direction in which the sheet is straight continues to infinity or, at least, to the edge of the sheet. The shape of a fluttering flag is often used as an example of a developable surface since a flag, it would seem, can be flattened out without distortion. However, a fluttering flag is clearly not a ruled surface since it is not developable, as seen in the following figure. Therefore, modeling it as a developable surface will inevitably fail to capture some of its properties.

Finally, note that not all ruled surfaces are developable as evidenced by the parabolic hyperboloid in the following figure.

In our pursuit of understanding curvature, one of our overarching goals will be to derive analytical criteria for developable surfaces.

Non-developable surfaces represent the third, highest "degree" of curvature. Such surfaces can be considered to be curved in a more profound way since they cannot be flattened without distortion. For example, a sphere -- or any part, no matter how small, thereof -- cannot be unwrapped onto a region of a plane without distortion. This is a well-known phenomenon in cartography when one attempts to represent the round Earth on a two dimensional map. When one attempts to do so in a smooth fashion, significant distortion results. The amount of distortion can be reduced with the help of tears, but it cannot be eliminated.

It follows from these observations that a non-developable surface does not admit an intrinsically regular coordinate system. This has important implications since a number of arguments used in Introduction to Tensor Calculus relied on the availability of affine coordinates. In particular, the fact the Riemann-Christoffel tensor vanishes fundamentally relied on affine coordinates. We can therefore expect that the corresponding surface object continues to vanish for developable surfaces but not for non-developable surfaces. In fact, this holds the promise of yielding a criterion for developable surfaces which will be explored in Chapter 7.

Let us now summarize the key concepts and notation introduced in Introduction to Tensor Calculus.

The position vector $\mathbf{R}$ at a point $P$ in a Euclidean space is the vector that emanates from an arbitrary fixed point $O$, known as the origin, and terminates $P$.

Refer the Euclidean space to arbitrary coordinates $Z^{i}$. Then the position vector $\mathbf{R}$ becomes a function of the coordinates which, in a three-dimensional space reads We typically abbreviate such expressions by representing the enumerated arguments of functions by a single letter, i.e.

The covariant basis $\mathbf{Z}_{i}$ is given by the partial derivative The covariant metric tensor $Z_{ij}$ is obtained by pairwise dot products of the covariant basis vectors, i.e. The contravariant metric tensor $Z^{ij}$ is the matrix inverse of $Z_{ij}$, i.e. where $\delta_{k}^{i}$ is the Kronecker delta system. The contravariant basis $\mathbf{Z}^{i}$ is given by the contraction The covariant and the contravariant bases are mutually orthogonal, i.e. Also, the last property can serve as an alternative definition of the contravariant basis $\mathbf{Z}^{i}$. The determinant of the covariant metric tensor $Z_{ij}$ is denoted by $Z$ and its square root, $\sqrt{Z}$, is referred to as the volume element.

The permutation systems $e_{ijk}$ and $e^{ijk}$ have the following values The Levi-Civita symbols $\varepsilon_{ijk}$ and $\varepsilon^{ijk}$ are defined by the identities

The complete delta system $\delta_{rst}^{ijk}$ is given by or, equivalently, and can be expressed in terms of the Kronecker delta by the identity The partial delta system $\delta_{rs}^{ij}$ has the values delta_{rs}^{ij}=left{ begin{tabular} {ll} $\begin{array} {c} \\ \phantom{+} 1\text{, }\\ \\ \phantom{+} \end{array}$ & $\begin{array} {l} \\ \text{if the superscripts and the subscripts are identical sets}\\ \text{of distinct numbers related by an \textit{even} permutation}\\ \phantom{+} \end{array}$ $\begin{array} {c} -1\text{,}\\ \\ \phantom{+} \end{array}$ & $\begin{array} {l} \text{if the superscripts and the subscripts are identical sets}\\ \text{of distinct numbers related by an \textit{odd} permutation}\\ \phantom{+} \end{array}$ $\begin{array} {c} \phantom{+} 0\text{,}\\ \end{array}$ & $\begin{array} {c} \text{for all other combinations of indices.}\\ \end{array}$ end{tabular} right. end{equation} It satisfies the identity and can be obtained from the complete delta system $\delta_{rst}^{ijk}$ by a single contraction, i.e. Thus, we also have the frequently used identity

The metric tensors are used to implement the operations of index juggling. Contraction with the covariant metric tensor $Z_{ij}$ is captured by index lowering, e.g. Contraction with the contravariant metric tensor $Z^{ij}$ is captured by index raising, e.g. The two operations are the inverses of each other.

The Christoffel symbol $\Gamma_{jk}^{i}$ is given by or, equivalently, by The symbol $\Gamma_{jk}^{i}$ is sometimes referred to as the Christoffel symbol of the second kind. The Christoffel symbol of the first kind $\Gamma_{i,jk}$ is obtained by lowering the index $i$ on $\Gamma_{jk}^{i}$, i.e. Alternatively, $\Gamma_{i,jk}$ is given by The partial derivative of the covariant metric tensor $Z_{ij}$ with respect to $Z^{k}$ is given by the identity The derivative of the volume element $\sqrt{Z}$ is given by The divergence $\nabla_{i}U^{i}$ of a variant field $U^{i}$ is given by the Voss-Weyl formula Correspondingly, the Laplacian $\nabla_{i}\nabla^{i}U$ of a scalar field $U$ is given by

The Christoffel symbol vanishes in affine coordinates, i.e. Its nonzero elements in cylindrical coordinates $r,\theta,z$ are while in spherical coordinates $r.\theta,\varphi$ they are

For a variant $T_{j}^{i}$ with a representative collection of indices, the covariant derivative $\nabla_{k}$ is given by the identity

The cornerstone feature of the covariant derivative is its tensor property -- namely, that it produces tensor outputs for tensor inputs. Furthermore, the covariant derivative satisfies the sum rule, the product rule, commutes with contraction. It coincides with the partial derivative in affine coordinates. Finally, it is metrinilic with respect to all of the fundamental tensors in a Euclidean space, i.e. and Note that the metrinilic property follows easily by a combination of the tensor property and the fact that the covariant derivative coincides with the partial derivative in affine coordinates.

Covariant derivatives commute. In other words, the commutator $\nabla _{i}\nabla_{j}-\nabla_{j}\nabla_{i}$ vanishes for all inputs, i.e. Since it can be generally shown that for any $T^{m}$, we conclude that The combination on the left is a tensor known as the Riemann-Christoffel tensor $R_{\cdot mij}^{k}$, i.e. Thus the fact that covariant derivatives commute is equivalent to the statement that the Riemann-Christoffel tensor vanishes, i.e.

Note that the logic that leads to this conclusion that covariant derivatives commute (and therefore $R_{\cdot mij}^{k}=0$) fundamentally relies on the Euclidean nature of the space. The argument goes like this. A Euclidean space admits an affine coordinate system where the covariant derivative coincides with the partial derivative, thus Meanwhile, as we know from ordinary Calculus, partial derivatives commute, i.e. Therefore, in affine coordinates, the commutator $\left( \nabla_{i}\nabla _{j}-\nabla_{j}\nabla_{i}\right) T^{k}$ vanishes, i.e. However, since $\left( \nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i}\right) T^{k}$ is a tensor, it must vanish in all coordinate systems which completes the argument.

Note, that we can expect the same argument to work for developable surfaces since they admit regular affine-like coordinate systems.

Conclusion to follow.