Let us agree to accept the concept of a surface without a definition. The surrounding Euclidean space will be referred to as the ambient space. As a whole, a surface is characterized by its shape. Locally, the shape of a surface is described by its curvature which, as we have just stated, is the primary object of our study. Due to curvature, most surfaces cannot accommodate straight lines. In other words, surfaces do not possess the kind of straightness that underpinned the concept of a Euclidean space. In particular, we cannot discuss geometric vectors on surfaces, since a vector with its tail on the surface will likely not be contained within the surface.

To every point on a smooth surface, there corresponds a unique tangent plane -- another concept that we will agree to accept without a definition for now but will later give an analytical characterization that will agree with our intuition. A vector pointing in the unique direction orthogonal to the tangent plane is known as a normal vector. A normal vector of length \(1\) is known as a unit normal and is denoted by \(\mathbf{N}\). With the help of the dot product, the fact that \(\mathbf{N}\) is unit length is captured by the equation

We called \(\mathbf{N}\) a unit normal, with emphasis on a. The indefinite article is appropriate since there are two opposite unit normals at every point. The symbol \(\mathbf{N}\) can denote either one of the two unit normals. However, in most analyses, a specific one of the two normals is selected, either arbitrarily or according to some geometric, typically coordinate-free, criterion. In those situations, the phrase the unit normal \(\mathbf{N}\) is typically used, even if the final selection has not yet been made.

What makes normal direction unique is the fact that a two-dimensional surface embedded in a three-dimensional space trails the dimension of the ambient space by \(1\). Embedded objects whose dimension trails that of the ambient space by \(1\) are known as hypersurfaces. Another example of a hypersurface that we will describe in this Chapter is a planar curve, i.e. a curve embedded in a plane.

This is about all that we are able to say about surfaces from a purely geometric point of view. Further progress demands that we impose a coordinate system upon the surface.

In order to enumerate the points of a two-dimensional surface, we need two coordinates. The surface coordinates will be denoted by the symbols \(S^{1}\) and \(S^{2}\) or, collectively, \(S^{\alpha}\). We have switched to the Greek alphabet because the number of coordinates on the surface is different from that in the ambient space for which we will continue to use Latin indices. In the context of two-dimensional surfaces, all Greek indices will range from \(1\) to \(2\).

For a canonical example, consider the surface of a sphere of radius \(R\). Introduce the coordinates \(S^{1}=\theta\) and \(S^{2}=\varphi\) as illustrated in the following figure. To make sense of these coordinates, simply imagine spherical coordinates \(r,\theta,\varphi\) in the ambient space and think of the sphere is the coordinate surface corresponding to the fixed value of \(r=R\). Then the varying values of the remaining coordinates \(\theta\) and \(\varphi\) act as the surface coordinates \(S^{1}\) and \(S^{2}\).

Importantly, the shape of the surface has significant influence on the way in which coordinates may be assigned. In particular, we may not be able to achieve some desired regularity, as we did with affine coordinates in the Euclidean space. Although we ought to clarify what we mean by regular , it is nevertheless clear that the presence of curvature imposes some constraints on the coordinate system. This insight alerts us to the fact that one of the central conclusions that we reached for Euclidean spaces may not hold on surfaces. Namely, our ability to choose an affine coordinate system in a Euclidean space leads to the Riemann-Christoffel equation where \(R_{\cdot mij}^{k}\) is the Riemann-Christoffel tensor given by If we are able to build an analytical framework that parallels the one we constructed for Euclidean spaces, we can expect the analogue of the Riemann-Christoffel tensor will reveal to us something about curvature. I hope that the thrill of anticipation of a new discovery is beginning to set in.

We will now do first what we previously did nearly at the end of our Euclidean space narrative: define tensors. The concept of a tensor will apply to variants defined on the surface. Consequently, the term surface tensor is often used to describe them, although we will almost always prefer tensor for short. The definition of a tensor will not surprise you since it will be exactly analogous to that of a Euclidean tensor. Suppose that the unprimed and primed coordinates \(S^{\alpha }\) and \(S^{\alpha^{\prime}}\) are related by the identities Introduce the Jacobians \(J_{\alpha^{\prime}}^{\alpha}\) and \(J_{\alpha} ^{\alpha^{\prime}}\) associated with this coordinate transformation The two Jacobians are the matrix inverses of each other, i.e. For future reference, the second order Jacobians \(J_{\alpha^{\prime} \beta^{\prime}}^{\alpha}\) and \(J_{\alpha\beta}^{\alpha^{\prime}}\) are defined by

A variant \(T_{\beta}^{\alpha}\), with a representative collection of indices, defined on the surface is an (absolute) tensor with respect to coordinate changes on the surface if its primed and unprimed values are related by the identity More generally, it is a relative tensor of weight \(m\) if where \(J\) is the matrix representing \(J_{\alpha^{\prime}}^{\alpha}\). It is left as an exercise to show that surface tensors satisfy all of the familiar properties of Euclidean tensors. Namely, the tensor property is reflexive, symmetric, and transitive. Furthermore, surface tensors satisfy the sum, product, and contraction properties. Finally, the quotient theorem remains valid.

In this Section, we will continue following our Euclidean blueprint and introduce the covariant and the contravariant bases \(\mathbf{S}_{\alpha}\) and \(\mathbf{S}^{\alpha}\), the covariant and the contravariant metric tensors \(S_{\alpha\beta}\) and \(S^{\alpha\beta}\), the area element \(\sqrt{S}\), and the Levi-Civita symbols \(\varepsilon^{\alpha\beta}\) and \(\varepsilon_{\alpha\beta }\).

The position vector \(\mathbf{R}\) with an arbitrary origin \(O\) is defined in the entire Euclidean space. Naturally, the origin \(O\) need not be on the surface. The surface restriction of \(\mathbf{R}\), i.e. the values of \(\mathbf{R}\) at points on the surface, can be thought of as a function of the surface coordinates \(S^{\alpha}\), i.e. or, following our convention of representing the collection of all independent variables by a single letter,

Suppose we fix the value of one of the coordinates, say \(S^{2}\), and consider the function By definition, \(\mathbf{R}\left( \gamma\right) \) traces out the coordinate corresponding to the fixed value of \(S^{2}\) and varying \(S^{1}\). Therefore, as we recall from Chapter TBD of Introduction to Tensor Calculus, the derivative \(\mathbf{R}^{\prime}\left( \gamma\right) \) represents a tangent vector to that coordinate line. This insight will help us with the geometric intuition of the covariant basis \(\mathbf{S}_{\alpha}\) which we will now introduce.

Following the Euclidean blueprint, the covariant basis \(\mathbf{S}_{\alpha}\) at a given point \(P\) is constructed by differentiating the position vector function \(\mathbf{R}\left( S\right) \) with respect to each of the surface variables, i.e. It is left as an exercise to demonstrate that \(\mathbf{S}_{\alpha}\) is a tensor, i.e. newline

Since the partial derivative corresponds to the ordinary derivative \(\mathbf{R}^{\prime}\left( \gamma\right) \) of the function \(\mathbf{R}\left( \gamma,S^{2}\right) \), the covariant basis vector \(\mathbf{S}_{1}\) is tangential to the coordinate line corresponding to varying \(S^{1}\) and fixed \(S^{2}\). Similarly, \(\mathbf{S}_{2}\) is tangential to the coordinate line corresponding to varying \(S^{2}\) and fixed \(S^{1}\). Thus, both \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\) are tangential to the surface \(S\) and therefore represent a basis for the tangent plane at \(P\). Thus, any vector emanating from \(P\) that lies in the tangent plane, and no vector emanating from \(P\) that lies outside of the tangent plane, can be expressed in terms of \(\mathbf{S}_{\alpha}\).

For a vector \(\mathbf{U}\) in the tangent plane, the coefficients \(U^{\alpha}\) in the linear decomposition are referred to as the contravariant surface components of \(\mathbf{U}\). It is left as an exercise to show that \(U^{\alpha}\) form a contravariant surface tensor.

Earlier in this Chapter, we agreed to accept the concept of the tangent plane without a definition. However, we are now able to define the tangent plane at the point \(P\) as the plane spanned by \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\). Of course, we must make sure that the resulting plane is invariant under a change of surface coordinates. In other words, that all bases \(\mathbf{S}_{\alpha^{\prime}}\) in all coordinate systems \(S^{\alpha ^{\prime}}\) span one and the same plane. It is left as an exercise to show that this follows from the tensor property of \(\mathbf{S}_{\alpha}\).

Since the basis vectors \(\mathbf{S}_{\alpha}\) span the tangent plane, they are orthogonal to the unit normal \(\mathbf{N}\), i.e. Furthermore, recall the normalization condition The last two equations may be adopted as the definition of the unit normal \(\mathbf{N}\).

Observe that the above equations define \(\mathbf{N}\) to within direction. Indeed, if a vector \(\mathbf{N}\) satisfies these equations, then so does \(-\mathbf{N}\). One way to choose a unique normal is to stipulate that the vectors \(\mathbf{S}_{1}\), \(\mathbf{S}_{2}\), and \(\mathbf{N}\) form a positively-orientated set. In this approach, however, \(\mathbf{N}\) flips under any change of coordinates that is not orientation preserving. However, we would like to think of \(\mathbf{N}\) as an invariant and will therefore choose a unique \(\mathbf{N}\) according to other, coordinate-free, considerations.

Once again following the Euclidean blueprint, the covariant metric tensor \(S_{\alpha\beta}\) is defined as the pairwise dot products of the elements of the covariant basis, i.e. The covariant metric tensor is symmetric, i.e. and positive definite.

The area element is \(\sqrt{S}\), where \(S\) is the determinant of the matrix associated with \(S_{\alpha\beta}\). The determinant \(S\) is a relative tensor of weight \(2\). Therefore, the area element \(\sqrt{S}\) is a relative tensor of weight \(1\), albeit only with respect to orientation-preserving coordinate changes.

The dot product of two tangent vectors \(\mathbf{U}=U^{\alpha}\mathbf{S} _{\alpha}\) and \(\mathbf{V}=V^{\alpha}\mathbf{S}_{\alpha}\) is given by The length of a tangent vector \(\mathbf{U}\) is given by

The contravariant metric tensor \(S^{\alpha\beta}\) is the matrix inverse of \(S_{\alpha\beta}\), i.e. where the Kronecker delta \(\delta_{\beta}^{\alpha}\) is, of course, defined as

The metric tensors \(S_{\alpha\beta}\) and \(S^{\alpha\beta}\) can be used for index juggling in a way that is completely analogous to the Euclidean case. For example, raising the subscript on a variant \(T_{\beta}\) results in a variant \(T^{\alpha}\) with a superscript, i.e. Similarly, lowering the superscript on \(T^{\beta}\) results in a variant \(T_{\alpha}\) with a subscript, i.e.

The contravariant basis \(\mathbf{S}^{\alpha}\) is defined by the equation Of course, we recognize it simply as the operation of raising the subscript on the covariant basis \(\mathbf{S}_{\beta}\). It is left as an exercise to show that the vectors \(\mathbf{S}^{\alpha}\) are related to the covariant basis \(\mathbf{S}_{\beta}\) by the identity Furthermore, the covariant basis \(\mathbf{S}_{\alpha}\) can be obtained from the contravariant basis \(\mathbf{S}^{\beta}\) by lowering the index, i.e. The components \(U_{\alpha}\) of a tangent vector \(\mathbf{U}\) with respect to \(\mathbf{S}^{\alpha}\), i.e. are known as the covariant surface components of \(\mathbf{U}\). They indeed form a covariant tensor and are related to the contravariant components \(U^{\alpha}\) by index juggling, i.e. Furthermore, the contravariant components \(U^{\alpha}\) of a vector \(\mathbf{U}\) in the tangent plane are given by the dot product while the covariant components are given by

The definitions of the permutation systems \(e_{\alpha\beta}\) and \(e^{\alpha\beta}\) are precisely as described in Chapter TBD of Introduction to Tensor Calculus. Namely, Similarly, the delta system \(\delta_{\gamma\delta}^{\alpha\beta}\) is the tensor product of two permutation systems, i.e. Note that we will commonly make use of the identity In terms of the permutation systems, the determinant \(S\) of the covariant metric tensor \(S_{\alpha\beta}\) is given by the expression

The Levi-Civita symbols \(\varepsilon^{\alpha\beta}\) and \(\varepsilon _{\alpha\beta}\) are defined by the equations The Levi-Civita symbols are tensors with respect to orientation preserving coordinate changes. They can be used for a number of purposes including the definition of surface vorticity.

Recall that a vector \(\mathbf{U}\) in the tangent plane can be represented by a linear combination of the covariant basis vectors \(\mathbf{S}_{\alpha}\), i.e. where the components \(U^{\alpha}\) are given by Meanwhile, a vector \(\mathbf{U}\) that does not lie in the tangent plane cannot be represented by a linear combination of the vectors \(\mathbf{S}_{\alpha}\). However, following the adage that all feasible tensor combinations are worthwhile, let us investigate the geometric meaning of the combination In other words, we will investigate the geometric meaning of the vector Of course, \(\mathbf{P}\) cannot equal \(\mathbf{U}\) since \(\mathbf{P}\) lies in the tangent plane while \(\mathbf{U}\) does not. However, as we are about to show, \(\mathbf{P}\) is the vector closest to \(\mathbf{U}\) among all vectors that lie in the plane. In other words, \(\mathbf{P}\) is the orthogonal projection of \(\mathbf{U}\) onto the plane. Note that orthogonal projection was described in Chapter TBD of Introduction to Tensor Calculus.

In order to demonstrate that \(\mathbf{P}\) is the orthogonal projection of \(\mathbf{U}\), we must show that the difference \(\mathbf{U}-\mathbf{P}\) is orthogonal to the tangent plane. Orthogonality to the tangent plane is equivalent to orthogonality to each of the elements of the covariant basis \(\mathbf{S}_{\alpha}\) or contravariant basis \(\mathbf{S}^{\alpha}\). The contravariant basis is more convenient for the present purpose. By dotting \(\mathbf{U}-\mathbf{P}\) with \(\mathbf{S}^{\alpha}\), we find Thus, \(\mathbf{U}-\mathbf{P}\) is indeed orthogonal to \(\mathbf{S}_{\alpha}\) and therefore \(\mathbf{P}\) is indeed the orthogonal projection of \(\mathbf{U}\) onto the tangent plane.

Let us take a moment to admire the compactness of the formula Recall that we have already discussed the topic of the component space representation of orthogonal projection in Section TBD of Introduction to Tensor Calculus Operating in a pre-tensor-notation context, we derived the formula A careful inspection of this equation will reveal that it represents the same computational algorithm as its compact tensor counterpart.

Let us also admire the great universality of the formula For a vector \(\mathbf{U}\) that lies in the tangent plane, this formula yields its contravariant components. Meanwhile, for a vector \(\mathbf{U}\) that lies outside the tangent plane, it yields the contravariant components of its orthogonal projection onto the plane, i.e. the vector in the plane closest to \(\mathbf{U}\).

Let us now turn our attention to projection away from the tangent plane, i.e. projection onto the normal direction. By analogy with consider the quantity and the vector The vector \(\mathbf{Q}\) is the orthogonal projection of \(\mathbf{U}\) away from the surface if the difference \(\mathbf{U}-\mathbf{Q}\) is orthogonal to \(\mathbf{N}\). To prove this, observe that as we set out to show.

It is a geometrically obvious fact that a vector is the sum of its projections onto and away from the tangent plane. In other words, where parentheses are needed to prevent the meaningless combinations \(\mathbf{US}_{\alpha}\) and \(\mathbf{NN}\). It is left as an exercise to demonstrate this identity algebraically. In Chapter 3, this identity will find a particularly elegant expression in terms of the ambient components.

The first real and exciting difference between a Euclidean space and an embedded surface comes in the definition of the Christoffel symbol \(\Gamma_{\alpha\beta}^{\gamma}\). Recall the definition of the ambient Christoffel symbol \(\Gamma_{ij}^{k}\) in Chapter TBD of Introduction to Tensor Calculus: The analogous definition is not possible on an embedded surface, since the vectors \(\partial \mathbf{S}_{\alpha}/\partial S^{\beta}\) may not lie in the tangent plane and can therefore not be expressed by linear combinations of \(\mathbf{S}_{\alpha} \). This is a welcome development as it is the first instance of curvature making its presence felt. Since curvature is a "second derivative" phenomenon, it is not surprising that it manifests itself in the derivative of the covariant basis rather than the covariant basis itself.

At this point, we have two alternatives at our disposal for defining the Christoffel symbol. The first is to imitate the explicit Euclidean formula and thus to define \(\Gamma_{\alpha\beta}^{\gamma}\) by the equation The second alternative is to define the Christoffel in terms of the derivatives of the metric tensor. In the context of a Euclidean space, we showed that Later on, in the context of Riemannian spaces in Chapter TBD of Introduction to Tensor Calculus, we adopted the above equation as the definition of the Christoffel symbol. Imitating this approach, known as intrinsic, we can define the \(\Gamma_{\alpha\beta}^{\gamma}\) by the equation

Naturally, the intrinsic approach is more universal, as it can be extended to Riemannian spaces. However, since we are pursuing a more geometric approach, we will choose the first definition, i.e. The symbol \(\Gamma_{\alpha\beta}^{\gamma}\) is sometimes referred to as the Christoffel symbol of the second kind. It is left as an exercise to show that, \(\Gamma_{\alpha\beta}^{\gamma}\) is symmetric in its subscripts, i.e. satisfies the identity and transforms according to the rule under a change of surface coordinates.

The Christoffel symbol of the first kind, \(\Gamma_{\gamma,\alpha \beta}\), is obtained by lowering the superscript \(\gamma\), i.e. It is left as an exercise to show that and that

For a variant \(T_{\gamma}^{\beta}\) with a representative collection of indices the definition of \(\nabla_{\gamma}\) reads The flagship characteristic of \(\nabla_{\alpha}\) is the tensor property: if the input variant is a tensor, the output is also a tensor with an additional covariant order. The covariant surface derivative satisfies the product rule, and commutes with contraction.

In the case of the Euclidean space, we found that the covariant derivative \(\nabla_{i}\) kills all metrics, i.e. It is left as an exercise to show that all but the first two analogous identities hold for the surface covariant derivative \(\nabla_{\gamma}\), i.e.

This is an exciting moment as we turn our attention to an analysis that brings out curvature. Recall the metrinilic property of the ambient covariant derivative \(\nabla_{i}\) with respect to ambient basis \(\mathbf{Z}_{j}\), i.e. This property is easy to show since, by definition, \(\nabla_{i}\mathbf{Z}_{j}\) is given by and vanishes since the Christoffel symbol is given by

The same argument is not available on a surface. The covariant derivative of the covariant basis is given by However, the expression on the right does not vanish since, as we discussed earlier, the combination \(\Gamma_{\alpha\beta}^{\omega}\mathbf{S}_{\omega}\) lies in the tangent plane while \(\partial\mathbf{S}_{\beta}/dS^{\alpha}\) may not -- due to curvature!

Nevertheless, the tensor \(\nabla_{\alpha}\mathbf{S}_{\beta}\) does have a special property -- each of its elements is orthogonal to the surface. In order to show this, recall that In order to take advantage of this relationship, dot both sides of the identity with \(\mathbf{S}^{\gamma}\): The first term on the right equals \(\Gamma_{\alpha\beta}^{\gamma}\). Meanwhile, for the second term, we have \(\Gamma_{\alpha\beta}^{\omega}\mathbf{S}^{\gamma }\cdot\mathbf{S}_{\omega}=\Gamma_{\alpha\beta}^{\omega}\delta_{\omega} ^{\gamma}=\Gamma_{\alpha\beta}^{\gamma}\). Thus the two terms cancel and we find as we set out to show.

The object \(\nabla_{\alpha}\mathbf{S}_{\beta}\) has one additional special property. Namely, it is symmetric, i.e. The proof of this identity is left as an exercise.

As we have already observed, curvature is the very reason why \(\nabla_{\alpha }\mathbf{S}_{\beta}\) does not vanish. Therefore, the object \(\nabla_{\alpha }\mathbf{S}_{\beta}\) holds the key to quantifying curvature. We will now exploit this insight by introducing the curvature tensor \(B_{\alpha\beta}\).

We have just established that each element in the tensor \(\nabla_{\alpha }\mathbf{S}_{\beta}\) is orthogonal to the surface. Thus, each element is proportional to the unit normal \(\mathbf{N}\). Denote by \(B_{\alpha\beta}\) the coefficients of proportionality between \(\nabla_{\alpha}\mathbf{S}_{\beta}\) and \(\mathbf{N}\), i.e. The object \(B_{\alpha\beta}\) is known as the curvature tensor. Its tensor property follows from the quotient theorem, as well as from the fact that it can be expressed explicitly in terms of tensor quantities. Namely, by dotting both sides of the above identity with the unit normal \(\mathbf{N}\), we find that Since \(\nabla_{\alpha}\mathbf{S}_{\beta}\) is symmetric, i.e. the curvature tensor, too, is symmetric, i.e. Raising the index \(\alpha\), we find As discussed in Chapter TBD of Introduction to Tensor Calculus, the system \(B_{\cdot\beta}^{\alpha}\) does not correspond to a symmetric matrix. Nevertheless, since the systems \(B_{\cdot\beta}^{\alpha}\) and \(B_{\beta}^{\cdot\alpha}\) are related by the above identity, we can omit the dot placeholder and write the mixed curvature tensor simply as \(B_{\beta }^{\alpha}\).

Note that in the identity the covariant derivative can be replaced with the partial derivative, i.e. This is so because and therefore Finally, since \(\mathbf{N}\) is orthogonal to \(\mathbf{S}_{\alpha}\), i.e. \(\mathbf{N}\cdot\mathbf{S}_{\gamma}=0\), we arrive at the desired result The advantage of this formula is that it eliminates the need for the Christoffel symbol and thus simplifies the calculation of the curvature tensor is some practical applications.

The invariant known as the mean curvature, is one of the most beautiful objects in our subject. Meanwhile, the determinant \(B\) of \(B_{\beta}^{\alpha}\), also an invariant, coincides with the Gaussian curvature as we described in the next Section. The vector \(\mathbf{N}B_{\alpha}^{\alpha}\) is known as the curvature normal, another term that we have encountered before -- namely, in Chapter TBD of Introduction to Tensor Calculus in the context of curves. The two definitions of the curvature normal will also be reconciled in the future.

Finally, notice one important aspect of the curvature tensor evident in both of the equations and Namely, its values depend on the choice of normal \(\mathbf{N}\). If the opposite choice is made, then the values of curvature tensor change their sign. Thus, the curvature tensor is defined with respect to a particular choice of normal, and when we state the values of the elements of the curvature tensor, we must specify which choice of normal it corresponds to. Of course, the same applies to the mean curvature \(B_{\alpha}^{\alpha}\). On the other hand, the Gaussian curvature, which is the determinant of \(B_{\beta}^{\alpha}\) is insensitive to choice of normal since multiplying a \(2\times2\) matrix by \(-1\) does not change its determinant. Similarly, the curvature normal \(\mathbf{N}B_{\alpha}^{\alpha}\) is insensitive to choice of normal since both terms in the product change sign when the choice of normal is reversed.

As we mentioned earlier, another manifestation of curvature is the loss of commutativity for the covariant derivatives. Recall that our proof of commutativity for the ambient covariant derivative \(\nabla_{i}\) rested on the availability of affine coordinates where the metric tensor \(Z_{ij}\) is constant from one point to another. Since we can no longer assume the availability of affine coordinates, we can no longer expect that the surface covariant derivatives commute. This insight opens a new avenue for the exploration of curvature. This avenue will be explored in Chapter 7. However, we will now mention some of the key landmarks from that Chapter.

Following the Euclidean blueprint, we can show that for a first-order variant \(T^{\gamma}\), the commutator \(\left( \nabla_{\alpha}\nabla_{\beta} -\nabla_{\beta}\nabla_{\alpha}\right) T^{\gamma}\) is given by where \(R_{\cdot\delta\alpha\beta}^{\gamma}\) is the surface Riemann-Christoffel tensor given by Since we cannot expect the surface covariant derivatives to commute, the Riemann-Christoffel tensor generally does not vanish. It is skew-symmetric in the first two indices, i.e. the last two indices, i.e. and is symmetric with respect to switching the sets of the first two and the last two indices, i.e. Owing to these symmetries, the Riemann-Christoffel symbol in a two-dimensional space can be expressed by the equation The invariant \(K\) is known as the Gaussian curvature. We have already encountered the concept of Gaussian curvature in the context of Riemannian spaces in Chapter TBD of Introduction to Tensor Calculus. Curved surfaces are thus breathing life into this concept and, indeed, that of a Riemannian space.

From the above equation, it follows immediately that \(K\) is given explicitly by the equation and, alternatively, by

The Riemann-Christoffel tensor is one of the central objects in the analysis of surfaces. One of the highlights of our entire narrative will be the Gauss equations which show that the Riemann-Christoffel tensor can be obtained from the curvature tensor. Since, as we demonstrated in Chapter TBD of Introduction to Tensor Calculus, the combination on the left is also given by where \(B\) is the determinant of the mixed curvature tensor \(B_{\beta}^{\alpha }\), we have and therefore the Gaussian curvature \(K\) coincides with \(B\), i.e. The profound importance of these identities will be discussed in Chapter 7.

We will now derive Weingarten's equation which is the formula for the covariant derivative of the unit normal. It reads Note, that since the unit normal \(\mathbf{N}\) is a variant of order zero, its covariant derivative coincides with its partial derivative, i.e.

It is not surprising to see the curvature tensor on the right side of Weingarten's equation since it is curvature that is responsible for the variability in the unit normal \(\mathbf{N}\). It is also not surprising that the result, being a linear combination of the covariant basis vectors \(\mathbf{S}_{\beta}\), is in the tangent plane. After all, \(\mathbf{N}\) has a constant length and, as we discovered in Section TBD of Introduction to Tensor Calculus, constant length implies that the derivative is orthogonal to the vector itself.

Since the unit normal \(\mathbf{N}\) is defined implicitly by the identities our derivation of its covariant derivative will also be implicit. Let us start by applying the covariant derivatives to both sides of the identity By the product rule, Since the two terms on the left are equal, we find This proves what we have already anticipated, that \(\nabla_{\alpha}\mathbf{N}\) is orthogonal to \(\mathbf{N}\) and therefore lies in the tangent plane.

Differentiating the identity yields According to the equation the first term in the previous equation is precisely \(B_{\alpha\beta}\). Therefore, Raising the index \(\beta\) yields Recall that the contravariant component \(U^{\alpha}\) of a vector \(\mathbf{U}\) in the tangent plane is given by the dot product Thus, the equation tells us that the contravariant component of the vector \(\nabla_{\alpha }\mathbf{N}\) is \(-B_{\alpha}^{\beta}\). In other words, which is precisely Weingarten's equation.

For a scalar field \(F\) defined on the surface, the vector is referred to as the surface gradient. To highlight its invariant nature, it may be denoted by the symbol \(\mathbf{\nabla}_{S}\), i.e. although we will, of course, prefer the indicial form. Much like its ambient counterpart, the surface gradient points in the direction of the greatest increase in \(F\) within the surface.

For a surface variant \(T^{\alpha}\), the combination is known as the surface divergence. By the Voss-Weyl formula, it is given by

The differential operator sometimes denoted by the invariant symbol \(\Delta_{S}\), is known as the surface Laplacian, the Laplace-Beltrami operator, or simply the Beltrami operator. It can be applied to a vector or a scalar field. An interesting relationship that features the surface Laplacian applied to the position vector is Its proof is left as an exercise.

By the Voss-Weyl formula, the surface Laplacian of a field \(F\) is given by

For a two-dimensional surface embedded in a three-dimensional Euclidean space, the concepts of the unit normal \(\mathbf{N}\) and therefore that of the curvature tensor \(B_{\beta}^{\alpha}\) rely on the fact that the surface is a hypersurface, i.e. its dimension trails that of the ambient space by \(1\). A curve embedded in a Euclidean plane, known as a planar curve, is also a hypersurface. Therefore, much of what we have already said about surfaces can be extended to planar curves essentially without change.

Since curves are one-dimensional objects, Greek indices assume a single value of \(1\). Therefore, let us repeat what we have already said in Section TBD of Introduction to Tensor Calculus. It may seem counterintuitive to use an index that assumes a single value. You may think that it would be easier to denote the coordinate by \(S^{1}\), rather than \(S^{\alpha}\), or to even drop the index altogether and denote it simply by \(S\). On the other hand, keep in mind that the indicial signature tells us how the object transforms under a change of coordinates. Therefore, preserving the indicial signatures is essential. Furthermore, indicial signatures inform us on how to combine variants together to produce other meaningful variants. Finally, preserving the indicial signatures will allow us to fit the theory of curves within the broader framework of embedded surfaces. For all of these reasons, we will preserve the indicial signatures of all variants. Thus, in a way, in this Chapter, we are aiming to take advantage of what curves have in common with two-dimensional surfaces. By contrast, Chapter TBD of Introduction to Tensor Calculus and Chapter 8 of this book exploit their one-dimensional nature.

Let us now repeat the entire surface narrative for curves in minimal fashion and, along the way, point out what remains exactly the same and what requires slight changes.

At each point, a planar curve is characterized by a unique tangent line. Furthermore, there is a unique direction orthogonal to the curve, i.e. orthogonal to the tangent line. Therefore, there are two unit normals pointing in opposite directions. The symbol \(\mathbf{N}\) represents the unit normal, in the sense that one of the two unit normals is chosen arbitrarily.

The covariant basis \(\mathbf{S}_{\alpha}\), defined by the equation consists of a single vector that points in the tangential direction. The covariant metric tensor consists of a single element that equals length squared of the basis vector \(\mathbf{S}_{\alpha}\). The line element \(\sqrt{S}\) equals the length of the basis vector.

The contravariant metric tensor \(S^{\alpha\beta}\) can still be defined by the identity Of course, in actuality, its only element is the reciprocal of the length squared of \(\mathbf{S}_{\alpha}\). The contravariant basis vector \(\mathbf{S}^{\alpha}\) is given by It points in the exact same direction as \(\mathbf{S}_{\alpha}\) and its length equals the reciprocal of the length of \(\mathbf{S}_{\alpha}\).

The permutation systems \(e_{\alpha}\) and \(e^{\alpha}\) each have one index and a single entry that equals \(1\). The Levi-Civita symbols \(\varepsilon_{\alpha}\) and \(\varepsilon^{\alpha}\) are defined by the equations and each has a single element: \(\varepsilon_{1}=\sqrt{S}\) and \(\varepsilon ^{1}=1/\sqrt{S}\). The Levi-Civita symbols are tensors with respect to orientation-preserving coordinate changes.

The entire machinery of Tensor Calculus continues to work. An unusual invariant not available in higher dimensions is \(\varepsilon^{\alpha }\mathbf{S}_{\alpha}\). It corresponds to the unit tangent vector \(\mathbf{T}\) that points in the same direction as \(\mathbf{S}_{\alpha}\). Much like \(\varepsilon^{\alpha}\), \(\mathbf{T}\) is an invariant only with respect to orientation-preserving coordinate changes. Indeed, we know that \(\mathbf{T}\) changes the direction when the orientation of the parameterization of the surface is reversed. Thus, its not an invariant in the full tensorial sense: the orientation-preserving stipulation is necessary.

This is a good moment to draw your attention once again to the elegance of Tensor Calculus. In Chapter TBD of Introduction to Tensor Calculus, the unit tangent \(\mathbf{T}\) was introduced as the derivative \(\mathbf{R} ^{\prime}\left( s\right) \) of the position vector \(\mathbf{R}\) with respect to the arc length. If the curve is referred to any other parameter \(\gamma\) then, in the absence of the tensor framework, the only way of arriving at \(\mathbf{T}\) is to divide \(\mathbf{R}^{\prime}\left( \gamma\right) \) by its length: Now, compare the above calculation to the tensor alternative I feel very strongly that the tensor expression is more elegant.

For the Christoffel symbol \(\Gamma_{\alpha\beta}^{\gamma}\), we once again use the geometric definition Of course, it has only a single element \(\Gamma_{11}^{1}\). The value of this element can be determined from the equation which follows from the definition. Let \(L\left( S\right) \) denotes the length of the covariant basis vector \(\mathbf{S}_{\alpha}\) as a function of the coordinate \(S^{\alpha}\), i.e. \(L=\sqrt{S}\), then the single element of the Christoffel symbol equals

The Riemann-Christoffel tensor \(R_{\cdot\delta\alpha\beta}^{\gamma}\) is given by However, in contrast with two-dimensional surfaces, the Riemann-Christoffel tensor on a one-dimensional curve does vanish, i.e. This follows from the availability of a perfectly regular coordinate system that we utilized so effectively in Chapter TBD of Introduction to Tensor Calculus -- namely, the arc length \(s\). When the curve is related to the arc length \(s\), i.e. \(S^{\alpha}=s\), the resulting covariant basis vector \(\mathbf{S}_{\alpha}\) is length \(1\) at all points. As a result, the covariant metric tensor \(S_{\alpha\beta}\) has the constant value of \(1\). Consequently, the Christoffel symbol vanishes identically and, with it, so does the Riemann-Christoffel tensor. An important consequence of this insight is the fact that covariant derivatives commute, i.e. on curves.

In Chapter 7, we will discover that this is a special case of a more general fact: the Riemann-Christoffel tensor vanishes on all surfaces that can be "straightened out" isometrically, i.e. without altering distances between points. Any two-dimensional surface, that can be made out of a sheet of paper by gently curving it without stretching or shrinking, i.e. isometrically, has this property. Such surfaces can just as easily be straightened back out isometrically. Special surfaces that have this property include cylinders and cones. Similarly, any curve that can be formed out of a string without stretching or shrinking, i.e. isometrically, has this property, as well. But that, of course, is all curves.

The curvature tensor \(B_{\alpha\beta}\) is defined in the same way as for two-dimensional surface: In the context of curves embedded in the plane, the mean curvature \(B_{\alpha}^{\alpha}\) may be referred to simply as curvature. The vector \(\mathbf{N}B_{\alpha}^{\alpha}\) is the curvature normal.

Note that the new definitions of the curvature and the curvature normal are in exact agreement with the concepts of the signed curvature \(\kappa\) and the curvature normal \(\mathbf{B}\) that were introduced in Chapter TBD of Introduction to Tensor Calculus, where the entire analysis was based on parameterizing the curve by its arc length \(s\). Despite the different approaches, the equivalence between the old and the new definitions is made obvious by the Tensor Calculus framework. The objects \(B_{\alpha}^{\alpha}\) and \(\mathbf{N}B_{\alpha} ^{\alpha}\) are invariants and therefore yield the same value regardless of the chosen parameterization. At the same time, under the arc-length parameterization, i.e. \(S_{\alpha\beta},S^{\alpha\beta}\equiv1\), the curvature normal \(\mathbf{N}B_{\alpha}^{\alpha}=\nabla_{\alpha} \nabla^{\alpha}\mathbf{R}\) becomes which coincides with Naturally, the same argument proves that the mean curvature \(B_{\alpha }^{\alpha}\) and the signed curvature \(\kappa\) are one and the same thing.

Exercise 2.1Show that surface tensors satisfy the sum, the product, and the contraction properties.

Exercise 2.2Show that surface tensors satisfy the quotient theorem.

Exercise 2.3Show that the surface tensor property is reflexive, symmetric, and transitive.

Exercise 2.4Show that

Exercise 2.5Show that

Exercise 2.6Show that the covariant components \(U_{\alpha}\) of a vector \(\mathbf{U}\) in the tangent space are given by

Exercise 2.7Demonstrate the equation algebraically. To this end, note that the set of vectors \(\mathbf{S}_{1}\), \(\mathbf{S}_{2}\), and \(\mathbf{N}\) represents a basis for the three-dimensional space and show that the two vectors on both sides of the equation produce the same values when dotted with each element of the basis.

Exercise 2.8Show that

Exercise 2.9Show that implies that

Exercise 2.10Show that

Exercise 2.11Show that \(\nabla_{\gamma}\) is metrinilic with respect to the covariant metric tensor \(S_{\alpha\beta}\), i.e.

Exercise 2.12Show that \(\nabla_{\gamma}\) is metrinilic with respect to the Kronecker delta \(\delta_{\beta}^{a}\), i.e.

Exercise 2.13Show that \(\nabla_{\gamma}\) is metrinilic with respect to the contravariant metric tensor \(S^{\alpha\beta}\), i.e.

Exercise 2.14Show that \(\nabla_{\gamma}\) is metrinilic with respect to the Levi-Civita symbols, i.e. and

Exercise 2.15Show the symmetry of the object \(\nabla_{\alpha}\mathbf{S}_{\beta}\), i.e.

Exercise 2.16Show that and These identities will prove useful on a few occasions.

Exercise 2.17The Riemann-Christoffel tensor \(R_{\gamma\delta\alpha\beta}\) with all subscripts is given by \(R_{\gamma\delta\alpha\beta}=S_{\gamma\omega} R_{\cdot\delta\alpha\beta}^{\omega}\). Show that

Exercise 2.18Show that the covariant derivatives commute when applied to a variant of order zero, i.e.

Exercise 2.19Show that the Laplacian of the position vector \(\mathbf{R}\) is given by

Exercise 2.20Construct an alternative narrative where Weingarten's equation is adopted as the definition of the curvature tensor \(B_{\alpha}^{\beta}\) from which the equation follows as a corollary.

Exercise 2.21For a two-dimensional surface, show that if \(\mathbf{N}\) is chosen so that the set \(\mathbf{S}_{1},\mathbf{S}_{2},\mathbf{N}\) is positively oriented, then

Exercise 2.22For a two-dimensional surface, show that the normal \(\mathbf{N}\) is given by the identity and that the resulting normal \(\mathbf{N}\) is such that the set \(\mathbf{S} _{1},\mathbf{S}_{2},\mathbf{N}\) is positively oriented. Of course, the above identity is simply the elegant tensor version of the formula found in elementary textbooks.

Over the next three exercises, we will derive the celebrated Gauss equations of the surface along with the equally elegant Codazzi equations

begin{exercise} Show that for the commutator \(\left( \nabla_{\alpha}\nabla_{\beta} -\nabla_{\beta}\nabla_{\alpha}\right) \) applied to a surface covariant tensor \(T_{\gamma}\), we have In particular, end{exercise}

Exercise 2.23Show that Thus,

Exercise 2.24From the above equation, derive the Codazzi equations as well as the identity Finally, show that this identity is equivalent to the Gauss equations of the surface