Our ultimate goal is to give a definition of the surface covariant derivative $\nabla_{\gamma}$ for variants with arbitrary collections of surface and ambient indices. However, rather than postulate the definition and subsequently present its properties, we will show how one can arrive at the definition on their own.

To this end, let us start with a variant $T^{i}$ that features a single ambient index, and attempt to discover the proper form for by assuming that $\nabla_{\gamma}$ has two desirable properties: one, that it satisfies the product rule and, two, that it is metrinilic with respect to the ambient basis $\mathbf{Z}_{i}$. After all, these are the two properties that are needed in order to transfer analysis from a Euclidean space to the component space. Indeed, when the covariant derivative is applied to the combination $T^{i}\mathbf{Z}_{i}$, it is by the product rule that it splits into two, i.e. and it is by the metrinilic property that the second term vanishes, i.e. Thus, it is by combination of these two properties that the covariant basis $\mathbf{Z}_{i}$ cleanly "separates" from the rest of the terms and thus facilitates conversion to component form.

For a more concrete example, consider the conversion from By the combination of the product rule and the metrinilic property, we have and therefore Thus, follows by equating the components of matching vectors. Similarly, with the help of the product rule and the metrinilic property, we can transition from Thus, requiring these two properties is essential for preserving our analytical framework.

It turns out that requiring the product rule and the metrinilic property essentially determined the surface covariant derivative. Indeed, for a first-order surface variant $T^{i}$, consider the vector field Apply the prospective covariant derivative $\nabla_{\gamma}$ to both sides of this identity. Since $\mathbf{T}$ is a variant of order zero, its covariant derivative must coincide with its partial derivative, i.e. As for the contraction $T^{i}\mathbf{Z}_{i}$, we have By the presumed product rule and metrinilic property for $\nabla_{\gamma}$, and the well-established product rule for $\partial/\partial S^{\gamma}$, we find

In order to evaluate the derivative $\partial\mathbf{Z}_{i}\left( S\right) /\partial S^{\gamma}$, represent the function $\mathbf{Z}_{i}\left( S\right)$ as a composition of the ambient function $\mathbf{Z}_{i}\left( Z\right)$ and the equation of the surface $Z^{i}\left( S\right)$, i.e. Then, by the chain rule, we have Recall that Thus,

Picking up where we left off, now becomes Since $T^{i}Z_{\alpha}^{j}\Gamma_{ij}^{k}\mathbf{Z}_{k}=Z_{\alpha}^{k} \Gamma_{km}^{i}T^{m}\mathbf{Z}_{i}$, we find Finally, matching up the components, we arrive at the final identity It is left as an exercise to show that for a variant $T_{j}$ with a subscript, the corresponding expression is Importantly, note the similarity between these relationships and the ambient covariant derivative identities

We would like to reiterate that the above analysis was explorative in nature. We merely identified the properties that we wished the surface covariant derivative to have and, on the basis of those properties, derived the identity that $\nabla_{\alpha}T^{i}$ must satisfy. We will move forward by using our present experience to postulate a definition of the surface covariant derivative, for which we will then prove that the desired properties actually hold.

Let us now build on our experience with the ambient covariant derivative and the explorations in the previous Section to give the following definition for the surface covariant derivative $\nabla_{\gamma}$ for a variant $T_{j}^{i}$ with a representative collection of ambient indices: But why stop here? Recall that we already have a definition for the surface covariant derivative for variants with surface indices, i.e. Thus, we can roll the two definitions into one that applies to variants with arbitrary combinations of ambient and surface indices. For a variant $T_{j\beta}^{i\alpha}$ with a fully representative collection of indices, the definition reads As usual, this equation is interpreted as a four-part recipe for each type of index. Many of its applications rely on its properties rather than the literal definition itself. For example, as we demonstrated above, this is the case for the two objects of present interest, the shift tensor $Z_{\alpha}^{i}$ and the components $N^{i}$ of the unit normal. On the other hand, when the above definition is used in the literal sense, it is usually being applied to variants with a small number of indices.

The flagship property of the surface covariant derivative is that it produces tensor outputs for tensor inputs. More specifically, it produces tensors with one surface covariant order greater than the input. This property, known as the tensor property, is the cornerstone of the surface covariant derivative because it guarantees that its use produces geometrically meaningful objects in all combinations of coordinate systems. The demonstration of this property can be carried out according to the approach we used in Chapter TBD of Introduction to Tensor Calculus for the ambient covariant derivative $\nabla_{i}$. Generalizing that approach to the surface covariant derivative is left as an important technical exercise for the reader.

We will now describe all of the remaining key properties of the surface covariant derivative. We will begin with the chain rule -- a new property that $\nabla_{\gamma}$ does not share with its predecessors. The rest of the properties will be familiar from studying the ambient covariant derivative as well as the limited version of the surface covariant derivative introduced in Chapter 2.

The surface covariant derivative $\nabla_{\gamma}$ applies to surface variants, such as the components $N^{i}$ of the unit normal or the shift tensor $Z_{\alpha}^{i}$. However, the surface variant may well be the surface restriction of a variant defined in the broader ambient space. For example, $\nabla_{\gamma}$ can be applied to the position vector $\mathbf{R}$, the ambient covariant basis $\mathbf{Z}_{i}$, or any other ambient variant. Of course, such variants are also subject to the ambient covariant derivative $\nabla_{i}$ and the role of the chain rule is to relate the two derivatives.

Naturally, ambient variants can have only ambient indices. While the chain rule holds for variants with arbitrary ambient indicial signatures, we will demonstrate it by a variant $T_{j}^{i}$ with a representative collection of indices. The chain rule reads In other words, the surface covariant derivative is the "orthogonal projection" of the ambient covariant derivative.

The proof of the chain rule requires a straightforward application of the definition of the surface covariant derivative. Suppose that $T_{j}^{i}\left( Z\right)$ denotes the dependence of $T_{j}^{i}$ on the ambient coordinates, while $T_{j}^{i}\left( S\right)$ denotes the dependence of its surface restriction on the surface coordinates. The two functions are related by the equation where $Z^{i}\left( S\right)$ are the equations of the surface.

By definition, we have The partial derivative $\partial T_{j}^{i}\left( S\right) /\partial S^{\gamma}$ can be obtained by differentiating the identity with respect to $S^{\gamma}$ which yields Since $\partial Z^{k}\left( S\right) /\partial S^{\gamma}=Z_{\gamma}^{k}$, we have and, therefore, Upon factoring out $Z_{\gamma}^{k}$, we find Since the quantity in parentheses is precisely $\nabla_{k}T_{j}^{i}$, we have arrived at the identity which is indeed the relationship we set out to show.

Note the interesting correspondence between the chain rule and the directional derivative formula or This formula states that $dF/dl$ is the orthogonal projection of $\nabla_{k}F$ onto the ray $l$. Similarly, we can think of $\nabla_{\gamma}T_{j}^{i}$ as a "directional derivative" of $T_{j}^{i}$ along the tangent plane. Correspondingly, $\nabla_{\gamma}T_{j}^{i}$ is the orthogonal projection of $\nabla_{k}T_{j}^{i}$ onto the tangent plane.

One immediate consequence of the chain rule is the metrinilic property of the surface covariant derivative with respect to the ambient metrics. Indeed, in order to evaluate $\nabla_{\gamma}\mathbf{Z}_{i}$, note that and, since we conclude that Naturally, the same conclusion can be reached for all other surface metrics. In summary, we have

We will now enumerate all of the remaining properties of the surface covariant derivative starting with the complete description of the metrinilic property.

At the end of the previous Section, we demonstrated the metrinilic property of the surface covariant derivative with respect to the ambient metrics. Note, however, that the full-fledged version of the derivative introduced in this Chapter coincides with its predecessor described in Chapter 2 for objects with surface indices. Therefore, much like its predecessor, the full-fledged derivative is metrinilic with respect to the surface metrics, i.e. Recall, however, that, crucially, $\nabla_{\gamma}$ is not metrinilic with respect to the surface covariant basis $\mathbf{S}_{\alpha}$, which is one important aspect in which surfaces differ from Euclidean spaces.

The surface covariant derivative satisfies the product rule, also known as the Leibniz rule. For example, The product rule can be demonstrated in the same way as the corresponding property for the ambient covariant derivative which was described in Chapter TBD of Introduction to Tensor Calculus. It is left as an exercise for the reader.

The combination of the metrinilic property and the product rule implies that indices can be juggled freely "across" the surface covariant derivative. The corresponding property for the ambient derivative was discussed in Chapter TBD of Introduction to Tensor Calculus.

For ambient indices, we have while implies Similarly, for surface indices, we have while implies

In summary, we are able to juggle indices freely without any regard for the presence of the surface covariant derivative. Justifying this statement is left as an exercise for the reader.

The surface covariant derivative $\nabla_{\gamma}$ commutes with contraction of both surface and ambient indices. The corresponding property for the ambient derivative was discussed in Section TBD of Introduction to Tensor Calculus.

Consider the expressions and Each of these expressions can be interpreted in two different ways depending on the order in which the covariant derivative and contraction are applied. However, regardless of the interpretation, the expressions yield the same results. The proof of this property as left as an exercise for the reader.

The surface covariant derivatives commute when applied to variants with ambient indices, i.e. The assumption that the ambient space is Euclidean is essential for this property to hold. When the ambient space is not Euclidean, this commutative property would not hold. The proof of this property, which is based on deriving the identity where $R_{\cdot mrs}^{i}$ is the (vanishing) ambient Riemann-Christoffel tensor, is left as an exercise.

On the other hand, the surface covariant derivatives do not commute for variants with surfaces indices. We are already familiar with this property from Chapter 2 where we discussed the fact that where the surface Riemann-Christoffel tensor $R_{\cdot \delta\alpha\beta}^{\gamma}$ is given by Note that Chapter 7 in its entirety is devoted to the study of the Riemann-Christoffel tensor.

Having established the product rule and the metrinilic properties of the surface covariant derivative $\nabla_{\gamma}$, we have retroactively validated the arguments made at the top of this Chapter by which we converted the vector identities and into their component counterparts and Like its vector counterpart, the latter formula is known as Weingarten's equation. It is left as an exercise to derive this formula by differentiating the explicit expression for the components $N^{i}$, i.e.

Finally, note that contracting both sides of the equation with $N_{i}$ gives an explicit expression for the curvature tensor $B_{\beta }^{\alpha}$, i.e. Similarly, contracting both sides of the equation with $Z_{i\gamma}$ (and, subsequently, renaming $\gamma$ into $\beta$) yields another explicit expression for the curvature tensor, i.e. When the ambient space is referred to affine coordinates, the covariant derivative reduces to the partial derivative, i.e.

Consider a scalar field $F$ defined in the ambient space. We can evaluate the directional derivative of $F$ along any direction at any point in the ambient space. However, at the points on the surface, there is a special direction that we might be particularly interested in -- the normal direction. The directional derivative in the normal direction is known as the normal derivative.

Recall from Chapter TBD of Introduction to Tensor Calculus, that the directional derivative $dF/dl$ of a field $F$ along the ray $l$ is given by the formula where $\mathbf{\nabla}F$ is the gradient of $F$ and $\mathbf{L}$ is the unit vector pointing in the direction of the ray $l$. Thus, at a point on a surface, the normal derivative of $F$, denoted by is given by which, in component form, reads

With the help of the normal derivative, we can decompose the ambient covariant derivative $\nabla_{i}F$ along the tangent and normal directions. Recall the projection formula Contracting both sides of this formula with $\nabla_{i}F$, we find Since we discover that Contracting both sides with the ambient contravariant basis $\mathbf{Z}^{k}$, we get the corresponding decomposition for the vector gradient $\mathbf{\nabla }F$, i.e. which can also be written in the invariant form

The above equation relates various first-order derivatives of $F$. We will now derive the beautiful invariant formula relating second-order derivatives of $F$. Note that each term in this formula has a straightforward geometric interpretation: $\nabla_{\alpha} \nabla^{\alpha}F$ is the surface Laplacian of $F$, $\nabla_{i}\nabla^{i}F$ is the ambient Laplacian of $F$, $N^{i}N^{j}\nabla_{i}\nabla_{j}F$ is the second-order normal derivative $\partial^{2}F/\partial n^{2}$ (this requires proof which is left as an exercise), and, finally, $B_{\alpha}^{\alpha} N^{i}\nabla_{i}F$ is the product of the mean curvature $B_{\alpha}^{\alpha}$ and the normal derivative $\partial F/\partial n$. In terms of the invariant symbols $\Delta_{S}$ for the surface Laplacian and $\Delta$ for the ambient Laplacian, the above identity can be written as

In order to prove this identity, recall that, by the chain rule and therefore Applying the product rule on the right, we find Since we have Applying the chain rule again to the second term, i.e. yields Finally, according to the projection formula we have Upon multiplying out the parentheses and absorbing the Kronecker delta, we arrive at the final result

This completes our discussion of the surface covariant derivative and puts us in a position to continue our exploration of curvature.

Exercise 5.1Show that for a first-order variant $T_{j}$ indexed by a subscript, the prospective surface covariant derivative $\nabla_{\gamma}$, subject to the product rule and the metrinilic property, must satisfy the identity

Exercise 5.2Use the techniques employed in Chapter TBD of Introduction to Tensor Calculus to demonstrate the tensor property of the surface covariant derivative.

Exercise 5.3Show that the surface covariant derivative satisfies the product rule. For example,

Exercise 5.4Justify each of the index juggling relationships in Section 5.4.3.

Exercise 5.5Show the contraction property of the surface covariant derivative.

Exercise 5.6Show that where $R_{\cdot mrs}^{j}$ is the ambient Riemann-Christoffel tensor. Since $R_{\cdot mrs}^{j}=0$, we can conclude that i.e. the surface covariant derivatives commute when applied to $T^{j}$.

Exercise 5.7Derive Weingarten's equation by applying the literal definition of the surface covariant derivative to the explicit expression for the components of the normal

Exercise 5.8Show that when the formula is used for the calculation of the curvature tensor, the term with the surface Christoffel symbol in $\nabla_{\alpha}Z_{\beta}^{i}$ can be omitted.

Exercise 5.9Show that the collection $\nabla_{\alpha}\nabla_{\beta}F$ of second-order surface covariant derivatives is related to the ambient covariant derivatives by the identity

Exercise 5.10Explain why the equation represents an invalid application of the chain rule.

Problem 5.1Show that See Problem TBD (second directional derivative) in Introduction to Tensor Calculus.