Our analytical exploration of curvature started with the realization that the spatial variability of the surface covariant basis $\mathbf{S}_{\alpha}$ comes from two sources: the potential non-uniformity of the surface coordinates $S^{\alpha}$ and the non-flatness -- in other words, curvature -- of the surface. While the non-uniformity of the surface coordinates can be effectively managed by switching from partial to covariant differentiation, the curvature of the surface is a matter of actual geometry. We therefore embrace it rather than seek to eliminate it from consideration by some analytical device.

The opportunity to quantify curvature presented itself when we discovered that each of the vectors in the tensor $\nabla_{\alpha}\mathbf{S}_{\beta}$ is orthogonal to the surface and therefore collinear with the unit normal $\mathbf{N}$. We then assembled corresponding coefficients of proportionality into a second-order system called the curvature tensor $B_{\alpha \beta}$, i.e. Dotting both sides of the above equation with $\mathbf{N}$ yields an explicit expression for $B_{\alpha\beta}$ captured by the equation Since the covariant basis $\mathbf{S}_{\beta}$ is the derivative of the position vector $\mathbf{R}$, i.e. the above identities may also be written in forms and in terms of the second derivatives of $\mathbf{R}$. Note that while, in general, the covariant derivatives $\nabla_{\alpha}$ and $\nabla_{\beta}$ do not commute, they do commute when applied to a variant of order zero. Indeed, for a zeroth-order variant $U$, we have Since the Christoffel symbol is symmetric in its subscripts, i.e. both of the terms in the expression for $\nabla_{\alpha}\nabla_{\beta}U$ are symmetric in $\alpha$ and $\beta$ and therefore Therefore $\nabla_{\alpha}\nabla_{\beta}\mathbf{R}$ is symmetric, i.e. In other words, from which it follows that the curvature tensor $B_{\alpha\beta}$ is symmetric, i.e. or, equivalently, Because of the symmetry of the curvature tensor, its mixed form can be written in this simplified form without the use of the dot placeholder. This subtle point was discussed in Chapter TBD of Introduction to Tensor Calculus. Finally, recall that, as we discussed in Chapter 2, the covariant derivative in the identity may be replaced with the partial derivative, i.e.

In Chapter 5, we observed that $B_{\alpha\beta}$ can be computed in the component space by the equation and the Weingarten equation where $Z_{\beta}^{i}$ is the shift tensor obtained by differentiating the equations of the surface $Z^{i}\left( S\right)$, i.e.

Importantly, the "sign" of the curvature tensor depends on the choice of normal. Recall that there are two unit normals pointing in opposite directions and, as we agreed in Chapter 2, the symbol $\mathbf{N}$ represents either one of them. Thus, depending on the choice of normal, the resulting elements in the tensor $B_{\alpha\beta}$ may summarily change sign. As a result, the curvature tensor inherits the same sign ambiguity of the unit normal. By extension, the same is true of the mean curvature $B_{\alpha }^{\alpha}$. The Gaussian curvature $K$, on the other hand, being the determinant of the $2\times2$ matrix corresponding to $B_{\beta}^{\alpha}$, is unchanged when the sign of the matrix is flipped and is therefore independent of the choice of normal. The product $\mathbf{N}B_{\alpha\beta}$, known as the vector curvature tensor, is also independent of the choice of normal since both elements in the product change sign when the normal is reversed.

This Section calls for a brief Linear Algebra preamble. As we described in Chapter TBD of Introduction to Tensor Calculus, a matrix $A$ representing a linear transformation corresponds to a mixed second-order tensor $A_{\cdot\beta}^{\alpha}$. Such a matrix $A$ is characterized by $n$ fundamental invariants -- its eigenvalues $\lambda_{1}$, $\lambda_{2}$, $\cdots$, $\lambda_{n}$. From these invariants, numerous others can be constructed. Two of the most noteworthy such invariants are the sum of the eigenvalues, which equals the trace of $A$, i.e. and their product, which equals the determinant, i.e. While the eigenvalues of a general $n\times n$ matrix may be difficult to calculate (in fact, no finite algorithm for calculating eigenvalues is possible for a matrix of dimension greater than $4$), the trace and the determinant are readily available.

For a self-adjoint transformation characterized by a symmetric tensor $A_{\cdot\beta}^{\alpha}$, i.e. all eigenvalues are real and the corresponding eigenvectors are orthogonal -- or, in the case of repeated eigenvalues, can be chosen to be orthogonal. The eigenvectors represent the critical points of the quadratic form subject to the unit-length constraint

When we transfer these facts onto the curvature tensor $B_{\beta}^{\alpha}$, we find that each of the objects mentioned above are of great geometric importance. The eigenvalues $\kappa_{1}$ and $\kappa_{2}$ are known as the principal curvatures while the corresponding eigenvectors are known as the principal directions. The sum $\kappa_{1}+\kappa_{2}$ is, of course, the mean curvature $B_{\alpha}^{\alpha}$ while their product $\kappa_{1}\kappa_{2}$ is, of course, the Gaussian curvature. We will now take a detailed look into each of these elements, starting with mean curvature.

The invariant is known as the mean curvature of the surface. The mean curvature is often denoted by the letter $\kappa$ but we will almost always prefer the symbol $B_{\alpha}^{\alpha}$.

The mean curvature is a particularly important invariant that appears in many fundamental identities. It is, by virtue of its ubiquity, the star of our entire narrative. Many physical phenomena associated with capillary pressure (i.e. the pressure differential across an interface between two fluids) are described by equations that invariably feature the mean curvature. Thus, not surprisingly, it plays an important role in the dynamics of fluid films. Furthermore, mean curvature plays an important role in the study of minimal surfaces, i.e. surfaces with a given boundary that have the least possible area. As we will demonstrate in a future book on the Calculus of Moving Surfaces, minimal surfaces are characterized by zero mean curvature.

The vector $\mathbf{N}B_{\alpha}^{\alpha}$ can be referred to as the vector mean curvature or, by analogy with curves that we will discover in Chapter 8, as the curvature normal. As we learned in Chapter 2, it equals the surface divergence of the covariant basis $\mathbf{S}^{\alpha}$ or the surface Laplacian of the position vector $\mathbf{R}$, i.e. Thus, an alternative expression for the mean curvature is The advantage of these identities is, of course, their pure geometric nature. For example, the first of the two equations will readily explain why the integral of the curvature normal over a closed surface is zero.

The determinant $B$ of the mixed curvature tensor $B_{\beta}^{\alpha}$ of a two-dimensional surface is the Gaussian curvature $K$. However, this is not the definition of Gaussian curvature but rather a consequence of the Gauss equations which we mentioned in Chapter 2 and will describe in detail in Chapter 7. As we stated in Chapter 2, by definition, the Gaussian curvature is the invariant $K$ in the two-dimensional identity where $R_{\alpha\beta\gamma\delta}$ is the Riemann-Christoffel tensor. Meanwhile, as we showed in Chapter TBD of Introduction to Tensor Calculus, any second-order system $B_{\beta}^{\alpha}$ in a two-dimensional space satisfies the equation Now, this is where the Gauss equations enter. From the Gauss equations we conclude that and therefore

We will save the rest of our discussion of Gaussian curvature for Chapter 7.

The eigenvalues of $B_{\beta}^{\alpha}$ are known as the principal curvatures. Thus, two-dimensional surfaces are characterized by two eigenvalues, $\kappa_{1}$ and $\kappa_{2}$. As mentioned earlier, the principal curvatures $\kappa_{1}$ and $\kappa_{2}$ correspond to the minimum and the maximum values of the quadratic form subject to the unit normalization condition Furthermore their sum is the mean curvature, i.e. and their product is the determinant $B$ of $B_{\beta}^{\alpha}$ which, by the Gauss equations, equals the Gaussian curvature $K$, i.e. The geometric interpretation of the principal curvatures $\kappa_{1}$ and $\kappa_{2}$ and the corresponding eigenvectors will be described below in a Section devoted entirely to that topic.

Note that some texts define the mean curvature as the average of the principal curvatures, i.e. which explains the use of the word mean in mean curvature. Such a definition has some advantages over our definition. For example, as we demonstrate below, the mean curvature $B_{\alpha}^{\alpha}$ of a sphere of radius $R$ is $-2/R$ which, in $n$ dimensions, generalizes to $-n/R$. The definition based on the average, on the other hand, yields the value of $-1/R$ in all dimensions. Nevertheless, we will stick with our definition since, in numerous applications, including the Calculus of Moving Surfaces, it is $B_{\alpha}^{\alpha}$, rather than $B_{\alpha}^{\alpha}/2$, that is most frequently encountered.

The metric tensor $S_{\alpha\beta}$ is sometimes referred to as the first fundamental tensor of the surface while the curvature tensor $B_{\alpha\beta}$ is referred to as the second fundamental tensor. The third fundamental tensor $C_{\beta}^{\alpha}$, which arises frequently in applications, is the tensor $C_{\beta}^{\alpha}$ defined by the equation In other words, $C_{\beta}^{\alpha}$ corresponds to the matrix square of the curvature tensor. Note that the fundamental tensors often figure as binlinear forms, such as $S_{\alpha\beta}U^{\alpha}V^{\beta}$. For this reason, they can also be referred to as fundamental forms or groundforms.

It is left as an exercise to show that Note the similarity among the equations that can act as definitions of the first groundform $S_{\alpha\beta}$, the second groundform $B_{\alpha\beta}$, and the third groundform $C_{\alpha\beta}$, especially if we denote $\mathbf{S}_{\alpha}=\nabla_{\alpha}\mathbf{R}$ by $\mathbf{R}_{\alpha}$ and $\nabla_{\alpha}\mathbf{N}$ by $\mathbf{N}_{\alpha}$. Then we have

As we know from Linear Algebra, the eigenvalues of $C_{\beta}^{\alpha}$ are $\kappa_{1}^{2},\kappa_{2}^{2},\cdots,\kappa_{n}^{2}$. Therefore, the invariant $C_{\alpha}^{\alpha}$ is given by while the determinant $C$ of $C_{\beta}^{\alpha}$ is given by

We have already studied planar curves in Chapter TBD of Introduction to Tensor Caclulus, where our analysis was based on parameterizing the curve by the arc length $s$. We will now approach curves from a different angle -- that is, as a special one-dimensional case of an embedded surface. In other words, we will, to the extent possible, ignore their one-dimensional nature and focus on what the general theory of surfaces can tell us about them.

A planar curve is a curve embedded in a Euclidean space. Therefore, the dimension of a planar curve trails that of its ambient space by $1$, which makes it a hypersurface. As a hypersurface, it is characterized by a unique (within sign) unit normal $\mathbf{N}$. Thus, the concept of the curvature tensor applies in full force, as is the concept of mean curvature. The curvature tensor $B_{\alpha\beta}$ is still defined by the equation or, equivalently, while the mean curvature $B_{\alpha}^{\alpha}$ is given by or, equivalently, Compare this equation to the identity from Chapter TBD of Introduction to Tensor Caclulus, where $\kappa$ is signed curvature and $\mathbf{T}\left( s\right)$ is the unit tangent as a function of the arc length $s$. Since, as it is left as an exercise to show, $\nabla_{\alpha}\mathbf{S}^{\alpha}$ is equivalent to $\mathbf{T}^{\prime}\left( s\right)$, we observe that for planar curves, the concept of mean curvature is equivalent to that of signed curvature.

Below, we will calculate the curvature tensor as well as the mean curvature for general planar curves represented in Cartesian and polar coordinates.

The geometric interpretation of principal curvatures involves curves embedded in two-dimensional surfaces. This configuration creates a fascinating interplay among three spaces: the one-dimensional curve, the two-dimensional surface, and the three-dimensional ambient Euclidean space. This intricate arrangement will yield a great number of intriguing relationships and insights. Some of these will appear in this book, but a great many more are beyond our scope -- and many more still are yet to be discovered.

Note that we have not yet developed a theory of surfaces embedded in non-Euclidean spaces, which is what a curve embedded in a general two-dimensional surface represents. Such a theory will be developed in Chapter 9. Fortunately, in order to understand the geometric interpretation of principal curvatures, we will only need to consider planar curves which, as we saw in the previous Section, we already have under our belt.

A planar curve arises when a surface is cut by a plane, as illustrated in the following figure. Our analysis will focus on a single point $P$ which lies on the curve at the intersection of the surface and the cutting plane. Furthermore, we will assume that the cutting plane is orthogonal to the surface at the point $P$. In other words, the cutting plane contains within in it the surface normal $\mathbf{N}$ at $P$. This condition does not specify the plane uniquely as there is an entire family of such planes obtained by rotation about the straight line containing $\mathbf{N}$.

The curve at the intersection of the surface and the plane can be analyzed from three distinct points of view. First, it can be viewed as a curve embedded in the two-dimensional surface. In this context, the surface can be described as the ambient space with respect to the curve. Second, it can be viewed as a curve embedded in the overall three-dimensional Euclidean space. And finally, it can be viewed as a curve embedded in the two-dimensional Euclidean cutting plane. Naturally, it is the last embedding with which we will concern ourselves here.

The curve represents a hypersurface with the respect to the plane. It is therefore characterized by its signed curvature $\kappa$ which, as we observed in the previous section, is equivalent to its mean curvature in the context of its embedding in the plane. Naturally, $\kappa$ depends on two characteristics: the curvature characteristics of the ambient surface and the orientation of the cutting plane. In this Section, we will succeed in capturing these dependencies by the elegant equation where $T^{\alpha}$ are the surface components of the unit tangent $\mathbf{T}$ to the curve. Clearly, the curvature tensor $B_{\alpha\beta}$ captures the curvature characteristics of the surface while the tangent $\mathbf{T}$ captures the orientation of the surface. The reader is invited to attempt deriving this identity on their own before we lay out the details below, as it relies on the techniques that we have developed earlier.

Before we turn to the derivation of the above identity, let us make one important geometric observation that will prove relevant later. Consider the unit normal $\mathbf{\tilde{N}}$ to the curve when viewed as a hypersurface within the cutting plane. At the point $P$, and likely only at the point $P$, it is colinear with the surface normal $\mathbf{N}$. This is by construction: the surface normal $\mathbf{N}$ at the point $P$ is orthogonal to all curves in the surface that pass through $P$, including the one formed by the intersection of the surface and the plane. The same cannot be said for any other point on the curve since there is no guarantee that the surface normal is contained within the cutting plane.

Despite our present affinity for treating curves as a special case of surfaces, we will, for the remainder of this Section, return to the style of analysis based on parameterizing the curve by its arc length $s$, which we employed in Chapter TBD of Introduction to Tensor Calculus. As before, denote the function representing the position vector $\mathbf{R}$ at the points on the curve by $\mathbf{R}\left( s\right)$, i.e. Recall that the first derivative of $\mathbf{R}\left( s\right)$ is the unit tangent $\mathbf{T}$, i.e. while the second derivative is the curvature normal $\mathbf{B}$, i.e. (Note that the determinant $B$ of the surface curvature tensor $B_{\beta }^{\alpha}$ is unrelated to the magnitude of the vector $\mathbf{B}$.) Importantly, for all $s$, the vectors $\mathbf{T}\left( s\right)$ and $\mathbf{B}\left( s\right)$ both lie within the cutting plane. Meanwhile, the curvature normal $\mathbf{B}$ is collinear with the in-plane normal $\mathbf{\tilde{N}}$ to the curve which coincides with the surface normal $\mathbf{N}$ at the point $P$. Thus, at the point $P$, the signed curvature $\kappa$, given by the identity is also given by the identity

All but the last equation in the previous paragraph are characteristic of the curve with respect to its embedding in the cutting plane. Thus, in order to derive the equation we must find a meaningful way to engage the differential characteristic of the surface.

Suppose that the curve within the surface is given by the equations Then the function $\mathbf{R}\left( s\right)$ can be constructed by composing the function $\mathbf{R}\left( S\right)$, i.e. $\mathbf{R}$ as a function of the surface coordinates $S^{\alpha}$, with the equations of the curve $S^{\alpha}\left( s\right)$, i.e. As we always do after establishing a relationship such as above, we will differentiate both sides with respect to the independent variable, i.e. the arc length $s$. By the chain rule, we have On the left, we recognize $\mathbf{R}^{\prime}\left( s\right)$ as the unit tangent $\mathbf{T}$. On the right, the partial derivative $\partial \mathbf{R}\left( S\right) /\partial S^{\alpha}$ is, of course, the surface covariant basis $\mathbf{S}_{\alpha}$. In other words, we have obtained the identity which tells us that the derivatives of the equations of the curve $S^{\alpha}\left( s\right)$ can be interpreted as the surface coordinates $T^{\alpha}$ of $\mathbf{T}$. In this respect, the equation is similar to a number of identities we have derived in the past, such as in Chapter TBD of Introduction to Tensor Calculus.

In order to get to curvature, we must differentiate the identity a second time. By the product rule, As we recalled above, the derivative $\mathbf{T}^{\prime}\left( s\right)$ is the curvature normal $\mathbf{B}$. Meanwhile, on the right, the object of utmost interest is the derivative which captures the rate of change of the covariant surface basis $\mathbf{S}_{\alpha}$ along the curve. Similarly to $\mathbf{R}\left( s\right)$, the function $\mathbf{S}_{\alpha}\left( s\right)$ can be constructed by substituting the equations of the curve $S^{\alpha}=S^{\alpha }\left( s\right)$ into $\mathbf{S}_{\alpha}\left( S\right)$, i.e. Then, by the chain rule, we have The derivative $\partial\mathbf{S}_{\alpha}\left( S\right) /dS^{\beta}$ is where the curvature tensor makes its appearance. Since and by definition of the covariant derivative we find Thus, becomes and therefore becomes where we have dropped the arguments of the functions for the sake of conciseness. Combining normal and tangential terms, we find

On the one hand, the combination which represents the tangential component of the curvature normal $\mathbf{B}$, is quite intriguing as it reminds us of the $\delta/\delta t$-derivative introduced in Section TBD of Introduction to Tensor Calculus. It will also reemerge in the future as we study the concept of the geodesic curvature for curves embedded in surfaces. On the other hand, since the curvature normal $\mathbf{B}$ is colinear with the surface normal $\mathbf{N}$ at the point $P$ as we established above, we can conclude that the tangential component of $\mathbf{B}$ vanishes, i.e. Therefore, the curvature normal $\mathbf{B}$ is given by Finally, since we arrive at the identity In other words, as we set out to show.

The geometric interpretation of the principal curvatures $\kappa_{1}$ and $\kappa_{2}$ follows immediately from the identity Recall that the principal curvatures $\kappa_{1}$ and $\kappa_{2}$ of a surface are defined as the eigenvalues of $B_{\beta}^{\alpha}$. From Linear Algebra, we know that these values are precisely the least and greatest values of the quadratic form subject to the constraint Therefore, the principal curvatures correspond to the least and the greatest signed curvatures of curves formed by intersecting the surface with orthogonal planes. Such values always exist and the corresponding directions, characterized by the unit tangents $\mathbf{T}_{1}$ and $\mathbf{T}_{2}$ are orthogonal. The following figure shows the two cutting planes corresponding to the principal directions. The following figure shows a point where the interplay between the principal curvatures is more interesting -- one is positive and the other is negative.

This interpretation of the principal curvatures immediately tells us the curvature invariants on the sphere of radius $R$. A normal plane cuts the sphere in a great circle of radius $R$. Thus, with respect to the outward normal, for both principal curvatures, we have Thus, the mean curvature of the sphere with radius $R$ with respect to the outward normal is given by The Gaussian curvature $K$, which is independent of the choice of normal, is given by

In this Section, we will expand the analyses presented in Chapter 4 to include the curvature tensor. One of the more convenient ways to calculate the curvature tensor by hand is represented by the form of Weingarten's equation. When the ambient space is referred to Cartesian coordinates, $\nabla_{\alpha}N_{i}$ coincides with the partial derivative $\partial N_{i}/\partial S^{\alpha}$ and the expression for the curvature tensor $B_{\alpha\beta}$ becomes This identity corresponds to a product of two matrices which, leveraging the notation introduced in Chapter 4, reads

For a sphere of radius $R$ referred to coordinates $\theta$ and $\varphi$ embedded in a Euclidean space referred to Cartesian coordinates, recall the following values: Note that the components $N^{i}$ in the above table correspond to the outward normal.

Differentiating the components of the normal with respect to the surface coordinates, we find Therefore, for the curvature tensor $B_{\alpha\beta}$, we have Raising the index $\alpha$ and, subsequently, $\beta$, we find

From the expression for $B_{\beta}^{\alpha}$, we can confirm our earlier calculation of the mean curvature $B_{\alpha}^{\alpha}$, i.e. Finally, the principal curvatures $\kappa_{1}$ and $\kappa_{2}$ are given by Note that the values of the curvature tensors, the mean curvature, and the principal curvatures correspond to the outward normal and would have the opposite sign had the other normal been selected.

Recall that Thus, and therefore the various forms of the curvature tensors are as follows: From $B_{\beta}^{\alpha}$, we see that the mean curvature $B_{\alpha}^{\alpha }$ is given by Finally, the principal curvatures are given by

Recall that Thus, As a result, the various forms of the curvature tensor are From $B_{\beta}^{\alpha}$, we see that the mean curvature $B_{\alpha}^{\alpha }$ is given by Finally, the principal curvatures are given by

Consider a surface of revolution given by the functions $G\left( \gamma\right)$ and $H\left( \gamma\right)$ as described in Sections 3.2.3 and 4.3, and let $G_{\gamma}$, $G_{\gamma\gamma}$, $H_{\gamma}$, and $H_{\gamma\gamma}$ denote the derivatives of the functions $G$ and $H$. Recall that We have As a result, the various forms of the curvature tensor are Therefore, the mean curvature $B_{\alpha}^{\alpha}$ is given by For the special case used very commonly in applications, we have Meanwhile, the mean curvature is given by and the principal curvatures are