Curves embedded in surfaces present us with a fascinating interplay among three spaces: the curve itself, its ambient surface, and the overall Euclidean space. There are also three embeddings to consider, i.e. that of the surface within the Euclidean space, of the curve within the Euclidean space, and, finally and most interestingly, of the curve within the surface. Although our interest is in all three embeddings and their interplay, it is the last embedding that is new to us and will therefore receive the greatest amount of attention.

Each embedding has a curvature tensor associated with it. The embedding of the surface within the Euclidean space is characterized by the now-familiar curvature tensor \(B_{\alpha\beta}\). The embedding of the curve within the Euclidean space is characterized by the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\) introduced in the previous Chapter. Finally, the embedding of the curve within the surface is characterized by the geodesic curvature tensor \(b_{\Phi\Psi}\), which will be introduced in this Chapter. The geodesic curvature tensor \(b_{\Phi\Psi}\) is entirely analogous to the curvature tensor \(B_{\alpha\beta}\) and the term geodesic simply refers to the fact that \(b_{\Phi\Psi}\) characterizes the embedding of the curve within the surface.

It is intuitively clear that the three curvature tensors are related. After all, if a surface is curved then it may not be able to accommodate straight curves, at least in some directions. Thus, high values in the curvature tensor \(B_{\alpha\beta}\) may induce relatively high values in the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\) depending on the orientation of the curve within the surface. This insight is reflected in the formula where \(S_{\Phi}^{\alpha}\) is the shift tensor of the curve with respect to the surface and \(\mathbf{n}\) is the unit normal to the curve within the surface. This formula is the culmination of this Chapter and may be interpreted as the attribution of the curve's overall curvature characteristics (captured \(\mathbf{B}_{\Phi\Psi}\)) to the curvature of the surface (captured by \(B_{\alpha\beta}\)) and the curvature of the curve within the surface (captured by \(b_{\Phi\Psi}\)).

Finally, we will continue to use the tensor notation with uppercase Greek letters for curve indices, despite the fact that curves are one-dimensional objects. In addition to the advantages mentioned in the previous Chapter, the analysis presented in this Chapter will benefit from this approach in one more important way. Namely, in higher-dimensional spaces, whether artificial Euclidean or Riemannian, one can consider similar curve-surface-space embeddings where the "curve", i.e. the innermost object, is of dimension greater than one. Thanks to our present use of the tensor notation to describe embedded curves, most of our analysis will generalize easily to those situations.

In order to describe a curve embedded in a surface which is, in turn, embedded in a Euclidean space we must introduce three coordinate systems: \(S^{\Phi}\) on the curve, \(S^{\alpha}\) on the surface, and \(Z^{i}\) in the ambient space. It is somewhat of a problem that the curve and surface coordinates are denoted by the same letter \(S\). For instance, the symbol \(S^{1}\) is ambiguous as it simultaneously stands for the curve coordinate and the first surface coordinate. However, we will typically refer to the coordinates collectively either by \(S^{\Phi}\) or by \(S^{\alpha}\) so this potential ambiguity will be avoided. As we are about to see, the only real problem momentarily arises in the vector equations of the surface and the curve.

Denote the dependence of the position vector \(\mathbf{R}\) on the ambient coordinates \(Z^{i}\) by the familiar function \(\mathbf{R}\left( Z\right) \). Let the vector equations of the surface be denoted by the function \(\mathbf{R}\left( S\right) \). Finally, the vector equations of the curve will also be denoted by \(\mathbf{R}\left( S\right) \) since both the curve and the surface coordinates are represented by the same letter \(S\). This is the momentary problem we have just alluded to in the previous paragraph. However, this problem is indeed only momentary since we never discuss the vector equations themselves but, rather, their derivatives with respect to the coordinates which produce the respective covariant bases \(\mathbf{Z}_{i}\), \(\mathbf{S}_{\alpha}\), and \(\mathbf{S}_{\Phi}\).

Those bases are given by familiar equations Note that in the last equation, the superscript of \(S^{\Phi}\) in the "denominator" makes it clear \(\mathbf{R}\left( S\right) \) in the "numerator" represents the vector equation of the curve.

Each of the objects above is well familiar to us from the earlier Chapters, along with the other fundamental objects that are constructed from them, such as the metric tensors, the contravariant bases, the volume, area, and length elements, the Levi-Civita symbols, the Christoffel symbols, and the Riemann-Christoffel tensors. Furthermore, we have also established relationships between the ambient and surface objects, as well as between the ambient and curve objects. However, we are yet to discuss the corresponding relationships between the surface and the curve objects, which is precisely the task to which we will now turn our attention.

The shape of a curve embedded in a surface can be described by the relationship between the curve coordinates \(S^{\Phi}\) and the surface coordinates \(S^{\alpha}\). The resulting equations are called the geodesic equations of the curve. In and of themselves, the geodesic equations of the curve do not carry the complete information about the overall shape of the curve in the three-dimensional space. In order to get the full picture, the geodesic equations need to be combined with the equation of the surface When these equations are composed, the result is, of course, the equations \(Z^{i}\left( S\right) \) of the curve in the three-dimensional space, i.e. This identity can be readily differentiated with respect to \(S^{\Phi}\) by applying the chain rule on the right. We have Of course, we are already familiar with the shift tensor that characterizes the embedding of the curve in the ambient space, as well as the shift tensor that characterizes the embedding of the surfaces in the ambient space. The only new element in the identity is object \(\partial S^{\alpha}\left( S\right) /\partial S^{\Phi}\) associated with the embedding of the curve within the surface. This object is denoted by \(S_{\Phi}^{\alpha}\), i.e. and is also referred to as a shift tensor, although it may also be referred to as the geodesic shift tensor.

The three shift tensors \(Z_{\Phi}^{i}\), \(Z_{\alpha}^{i}\), and \(S_{\Phi }^{\alpha}\) are related by the identity This identity does not have an official name, but it is well described as cascading projections. Recall from Chapter 3 that \(Z_{\alpha}^{i}\) represents projection from the ambient three-dimensional space onto the surface. Similarly, \(Z_{\Phi}^{i}\) represents projection from the ambient space onto the curve and \(S_{\Phi}^{\alpha}\) represents projection from the tangent plane onto the curve. The identity tells us that projection from the space onto the curve is equivalent to projection from the space onto the surface followed by projection onto the curve. Our derivation may have been straightforward, but this is actually a nontrivial geometric result.

Next, let us convert the relationship among the shift tensors into a relationship between the curve and surface covariant bases. To this end, contract both sides of the above with the ambient basis \(\mathbf{Z}_{i}\), i.e. Since \(Z_{\Phi}^{i}\mathbf{Z}_{i}=\mathbf{S}_{\Phi}\) and \(Z_{\alpha} ^{i}\mathbf{Z}_{i}=\mathbf{S}_{\alpha}\), we arrive at the relationship which, given our accumulated experience, we could have all but predicted. And we can also predict that a differentiation of this identity will yield a relationship among the various characteristics of curvature.

Meanwhile, the equation yields the relationship between the curve and the surface metric tensors, i.e. The equivalent forms of this identity are and

Since the embedded curve is a hypersurface with respect to the ambient surface, it has a well-defined, to within sign, unit normal \(\mathbf{n}\) that lies in the tangent plane to the surface. We will call \(\mathbf{n}\) the geodesic normal. It is specified uniquely, to within sign, by the condition that it lies in the tangent plane, is orthogonal to textbf{\(S\)}\(_{\Phi}\), i.e. and that it has unit length, i.e. Denoting the components of \(\mathbf{n}\) with respect to the surface basis \(\mathbf{S}_{\alpha}\) by \(n^{\alpha}\), i.e. the above conditions read and The explicit expression for \(n^{\alpha}\), analogous to its classical counterpart for the components of the surface normal \(\mathbf{N}\), reads Naturally, it exhibits the same limitations: the resulting variant may differ in sign from the components of an a priori choice of normal. Furthermore, the quantity on the right is a tensor only with respect to the orientation-preserving coordinate changes.

The geodesic normal \(\mathbf{n}\) can also be decomposed with respect to the ambient basis \(\mathbf{Z}_{i}\), i.e. Since the ambient components of a vector are obtained by contracting its surface components with the shift tensor \(Z_{\alpha}^{i}\), i.e. we have and therefore

The projection formula first derived in Chapter 2, demonstrates that a vector \(\mathbf{U}\) on the surface, can be represented as a sum of its tangential and normal projections, \(\left( \mathbf{S}^{\alpha}\cdot\mathbf{U}\right) \mathbf{S}_{\alpha}\) and \(\left( \mathbf{U}\cdot\mathbf{N}\right) \mathbf{N}\). In component form, the projection formula reads

Naturally, an analogous formula exists for curves embedded in surfaces. It features the curve covariant basis textbf{\(S\)}\(^{\Phi}\) and the geodesic normal \(\mathbf{n}\) and applies to vectors in the tangent plane. Specifically, for a vector \(\mathbf{u}\) in tangent plane, we have where the terms \(\left( \mathbf{S}^{\Phi}\cdot\mathbf{u}\right) \mathbf{S}_{\Phi}\) and \(\left( \mathbf{u}\cdot\mathbf{n}\right) \mathbf{n}\) are interpreted as the components of \(\mathbf{u}\) that are tangential and orthogonal to the curve. The component form of the same equation reads which is usually applied in the form whenever the combination \(S_{\Phi}^{\alpha}S_{\beta}^{\Phi}\) is encountered.

Similarly to the object \(Z_{\alpha}^{i}Z_{j}^{\alpha}\) discussed in Chapter 3, \(S_{\Phi}^{\alpha}S_{\beta}^{\Phi}\) represents orthogonal projection from the tangent space of the surface to the tangent space of the curve. Also recall from Chapter 8, that the shift tensor \(Z_{\Phi }^{i}\) is proportional to the ambient components \(T^{i}\) of the unit tangent \(\mathbf{T}\). In fact, Similarly, the shift tensor \(S_{\Phi}^{\alpha}\) is proportional to the surface components \(T^{\alpha}\) of \(\mathbf{T}\), where Since the combination \(T_{\alpha}T^{\beta}\) also represents orthogonal projection from the tangent space of the surface to the tangent space of the curve, we should expect that The simplest way to demonstrate this identity is to note that, similarly to the identity we have Thus, recalling that \(\varepsilon_{\Phi}\varepsilon^{\Psi}=\delta_{\Phi} ^{\Psi}\), we have as we set out to show.

We have already defined the curve covariant derivative \(\nabla_{\Phi}\) for objects and ambient and curve indices. We must now extend it to surface indices. This can be accomplished according to the well established blueprint. However, rather than give an overwhelming expression for a variant \(T_{j\beta\Psi}^{i\alpha\Phi}\) with a fully representative collection of indices, we will give three separate expressions for \(T_{j}^{i} \), \(T_{\beta}^{\alpha}\), \(T_{\Psi}^{\Phi}\) since the tactic of the definition is simply to indicate the proper treatment of each kind of index. We have:

By now, we have significant experience constructing differential operators that possess the tensor property along with a slew of other desirable characteristics. We will therefore limit ourselves to stating the definition and assume that the reader will fill in all of the necessary details.

For a variant \(T_{j}^{i}\) defined in the ambient Euclidean space, \(\nabla_{\Phi}\) satisfies the chain rule while for a variant \(T_{\beta}^{\alpha}\) defined in the wider ambient surface outside, \(\nabla_{\Theta}\) satisfies the chain rule As a result, \(\nabla_{\Theta}\) is metrinilic with respect to all of the ambient metrics, including the covariant and contravariant bases \(\mathbf{Z}_{i}\) and \(\mathbf{Z}^{i}\). However, while \(\nabla_{\Theta}\) is metrinilic with respect to the scalar surface metrics, we are not able to conclude the same with respect to the surface covariant and contravariant bases \(\mathbf{S}_{\alpha}\) and \(\mathbf{S}^{\alpha}\) due to the lack of the metrinilic property of the surface covariant derivative \(\nabla_{\alpha}\) with respect to those objects. Indeed, since we have, by the chain rule Let us now summarize the results of applying the curve covariant derivative to the ambient, surface, and curve bases: where \(\mathbf{B}_{\Phi\Psi}\) is the vector curvature tensor introduced in the preceding Chapter.

The above expressions bring us face to face with the analysis of curvature which is the task to which we now turn.

The purpose of geodesic curvature is to characterize the shape of an embedded curve relative to the ambient surface. This concept is exemplified by the difference between roads that we would differentiate as straight and curved.

Naturally, even the "straight" road in the picture above is not straight in the absolute sense since it curves with the curvature of the Earth. However, on an intuitive level, it is "as straight as possible" given that it is constrained to follow the surface of the Earth. We might describe it as straight relative to the surface of the Earth. Such roads also correspond to our intuitive understanding of what it takes to connect two towns by the shortest possible path. The road in the second picture, on the other hand, would be described as curved relative to the surface of the Earth. We will wish to design the geodesic curvature tensor so that it is low -- ideally, zero -- for the "straight" road and high for the other road. Notice, then, that it is its geodesic curvature that makes a road potentially more dangerous and thus necessitates a lower speed limit -- curvature corresponds to acceleration -- and a double yellow separator line.

Let us proceed by analogy with the classical curvature tensor \(B_{\alpha\beta }\) which characterizes a surface embedded in a Euclidean space. Recall its definition By dotting both sides with the surface normal \(\mathbf{N}\) we obtain an explicit expression for \(B_{\alpha\beta}\), i.e. An analogous definition for the geodesic curvature tensor \(b_{\Phi\Psi}\) would read However, this definition is flawed. Indeed, while the vector \(\mathbf{B} _{\Phi\Psi}=\nabla_{\Phi}\mathbf{S}_{\Psi}\) is orthogonal to the embedded curve, it is not necessarily found in the tangent plane and is therefore not necessarily proportional to the geodesic normal \(\mathbf{n}\). As a result, we will look to the equation for the analogy and define the geodesic curvature tensor \(b_{\Phi\Psi}\) by the equation or, equivalently The tensor \(b_{\Phi\Psi}\) thus defined is called the geodesic curvature tensor.

Observe, crucially, that the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\) is characteristic strictly of the curve's embedding in the overall Euclidean space and has nothing to do with the ambient surface. Thus, the dependence of the geodesic curvature tensor on the surface enters strictly through the normal \(\mathbf{n}\), i.e. the direction of the curve relative to the surface.

The invariant is known as the geodesic curvature although, perhaps, it would have been better described as the geodesic mean curvature. It can be expressed as the dot product of the geodesic normal \(\mathbf{n}\) and the curvature normal \(\mathbf{B}_{\Phi}^{\Phi}\), i.e. In other words, it is the projection of the curvature normal onto the geodesic normal. Thus, the geodesic curvature cannot exceed the absolute curvature \(\sigma\), i.e. Recall from Chapter 8, that \(\sigma\) is defined as the magnitude of the curvature normal \(\mathbf{B}_{\Phi}^{\Phi}\). Furthermore the geodesic curvature vanishes when the curvature normal \(\mathbf{B}_{\Phi}^{\Phi}\) is orthogonal to the geodesic normal \(\mathbf{n}\).

A great deal of geometric intuition about geodesic curvature and its relationship to the curvature normal can be gained by considering circles on the surface of a sphere. The following figure shows three parallel circles of different radii. For these circles, \(\mathbf{B}_{\Phi}^{\Phi}\) is easy to visualize as it lies in the same plane as the circle and has magnitude that is the reciprocal of the radius. Thus, for a small circle, which we would perceive as highly curved, \(\mathbf{B}_{\Phi}^{\Phi}\) is closely aligned with \(\mathbf{n}\). For a large circle \(\mathbf{B}_{\Phi}^{\Phi}\) is nearly orthogonal to \(\mathbf{n}\).

A great circle is a circle on the surface of the sphere with the greatest possible radius which, of course, equals the radius of the sphere itself. It has the property that its center is located at the center of the sphere. In other words, a great circle is the intersection of the sphere with a plane passing through its center. Consequently, its curvature normal \(\mathbf{B}_{\Phi}^{\Phi}\) is orthogonal to the surface of the sphere and is therefore orthogonal to \(\mathbf{n}\). As a result, the geodesic curvature \(b_{\Phi}^{\Phi}\) vanishes, i.e. \(b_{\Phi}^{\Phi}=0\). Note that it is intuitively clear that a "straight" road on the surface of the Earth follows a great circle.

In fact, let us determine the actual value of the geodesic curvature of a circle of radius \(r\) on the surface of a sphere of radius \(R\). This configuration is illustrated in the following figure. Since while we have or, equivalently, Note that this formula was derived without introducing any surface or ambient coordinates, even though the angle \(\theta\) is reminiscent of the longitudinal angle in spherical coordinates.

According to the above formula \(b_{\Phi}^{\Phi}\) vanishes when \(r=R\) which confirms our intuition that a great circle is straight relative to the sphere. Also, when \(r\) is small, we have which tells us that the geodesic curvature of a small circle essentially matches its absolute curvature and is independent of the radius of the sphere. Finally, note that the sign of the geodesic curvature depends on the a priori choice of the geodesic normal \(\mathbf{n}\). Had we chosen the opposite direction of \(\mathbf{n}\), all of the values of \(b_{\Phi}^{\Phi}\) would have had the opposite sign.

A shortcoming of the definition is that it involves a quantity -- namely, the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\) -- which is connected with the embedding of the curve in the overall three-dimensional Euclidean space. Meanwhile, the goal of the geodesic curvature tensor \(b_{\Phi\Psi}\) is, of course, to characterize the shape of the curve relative to the ambient surface. We would therefore like to obtain an expression for \(b_{\Phi\Psi}\) strictly in terms of the equations of the curve and their derivatives.

Let us think back to the classical curvature tensor \(B_{\alpha\beta}\) and its coordinate space expression The analogous equation for the geodesic curvature tensor reads however, at this point it is not clear whether this definition is equivalent to the one we have already made. In the next Section, we will show that it is indeed equivalent, and will also show the surprising identity It is surprising because, as we have already discussed, \(\nabla_{\Phi }\mathbf{S}_{\Psi}=\mathbf{n}b_{\Phi\Psi}\) does not hold.

The geodesic curvature \(b_{\Phi}^{\Phi}\) is given by the equation A more general form of this equation is

Let us go back to the analysis of the equation that defines the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\). Notice that we can engage the geometry of the surface by substituting the previously obtained expression for textbf{\(S\)}\(_{\Psi}\) in terms of the surface basis \(\mathbf{S}_{\alpha}\), i.e. By the product rule, the identity becomes As we discovered earlier, \(\nabla_{\Phi}\mathbf{S}_{\alpha}\) is given by the equation Thus, we have arrived at the powerful identity which represents a decomposition of the vector curvature tensor \(\mathbf{B} _{\Phi\Psi}\) into its normal and tangential components with respect to the surface. The normal component \(B_{\alpha\beta}S_{\Phi}^{\alpha}S_{\Psi} ^{\beta}\) will be analyzed shortly. At the present time, let us direct our attention to the tangential components \(\nabla_{\Phi}S_{\Psi}^{\alpha}\).

As we discussed in the previous Section, we hope that the quantity \(\nabla_{\Phi}S_{\Psi}^{\alpha}\) is related to the geodesic curvature tensor \(b_{\Phi\Psi}\). This is indeed easily seen to be the case by dotting both sides of the equation with the geodesic normal \(\mathbf{n}\), i.e. The dot product \(\mathbf{n}\cdot\mathbf{B}_{\Phi\Psi}\) on the left is the geodesic curvature tensor \(b_{\Phi\Psi}\). Additionally, \(\mathbf{n} \cdot\mathbf{N}=0\), and \(\mathbf{n}\cdot\mathbf{S}_{\alpha}=n_{\alpha}\). Thus, \(b_{\Phi\Psi}\) is given by as was our hope.

However, as we have already mentioned, we can demonstrate the stronger relationship Indeed, since in the equation both \(\mathbf{B}_{\Phi\Psi}\) and \(\mathbf{N}B_{\alpha\beta}S_{\Phi}^{\alpha }S_{\Psi}^{\beta}\) are orthogonal to the curve, we can conclude that \(\mathbf{S}_{\alpha}\nabla_{\Phi}S_{\Psi}^{\alpha}\) is, too, orthogonal to the curve. Since, at the same time, \(\mathbf{S}_{\alpha}\nabla_{\Phi}S_{\Psi }^{\alpha}\) lies in the tangent plane, it must be a scalar multiple of the geodesic normal \(\mathbf{n}\). In other words, for some \(c_{\Phi\Psi}\). Dotting both sides with \(\mathbf{n}\) proves that \(c_{\Phi\Psi}=b_{\Phi\Psi}\) and thus From this identity, it immediately follows that as we set out to show. A more algebraic proof of this identity can be found in one of the exercises.

With this newly obtained expression for \(\nabla_{\Phi}S_{\Psi}^{\alpha}\), the equation becomes Since \(\mathbf{S}_{\alpha}n^{\alpha}=\mathbf{n}\), we arrive at the beautiful identity which captures the interplay between the three curvature tensors \(\mathbf{B}_{\Phi\Psi}\), \(B_{\alpha\beta}\), and \(b_{\Phi\Psi}\). We will, therefore, refer to this identity as the equation of the three curvature tensors. To summarize the roles of these tensors one more time, the vector curvature tensor \(\mathbf{B}_{\Phi\Psi}\) characterizes the shape of the curve relative to the Euclidean space, the classical curvature tensor \(B_{\alpha\beta}\) characterizes the shape of the surface relative to the Euclidean space, and the geodesic curvature tensor \(b_{\Phi\Psi}\) characterizes the shape of the curve relative to the surface.

Raising the index \(\Phi\) in the equation and subsequently contracting on \(\Phi\) and \(\Psi\) (as well as juggling \(\alpha\)) yields the stunning formula that relates five invariants: the curvature normal \(\mathbf{B} =\mathbf{B}_{\Phi}^{\Phi}\), the invariant \(B_{\beta}^{\alpha}S_{\alpha}^{\Phi }S_{\Phi}^{\beta}\) known as the normal curvature and discussed below, the geodesic curvature \(b_{\Phi}^{\Phi}\), the surface normal \(\mathbf{N}\) and the geodesic normal \(\mathbf{n}\). It is fair to say that it is rare for a fundamental formula to relate as many as five invariants. Interestingly, we can increase that number to six if we replace the curvature normal \(\mathbf{B}\) with the product of the absolute curvature \(\sigma\) and the principal normal \(\mathbf{P}\), i.e.

The invariant in the equation is known as the normal curvature. Since the shift tensor \(S_{\Phi }^{\alpha}\) depends only on the direction of the curve within the surface, the normal curvature is dictated solely by the curvature characteristics of the ambient surface and the direction of the curve. Also, since the sign of the curvature tensor depends on the choice of the surface normal \(\mathbf{N}\), the same is true for the normal curvature.

There are two alternative expressions for the normal curvature that are free of curve indices. First, with the help of the geodesic projection formula we obtain an identity that features the mean curvature \(B_{\alpha}^{\alpha}\), i.e. Alternatively, with the help of the identity where \(T^{\alpha}\) are the components of the unit tangent \(\mathbf{T}\), we obtain an expression for the normal curvature that is a simple quadratic form in \(T^{\alpha}\). Recall from Chapter TBD in Introduction to Tensor Calculus, that the largest and the smallest values of \(B_{\beta}^{\alpha}T_{\alpha}T^{\beta}\) equal the eigenvalues of \(B_{\beta}^{\alpha}\), i.e. the principal curvatures \(\kappa_{1}\) and \(\kappa_{2}\).

We can now rewrite the equation of the three curvature tensors and the equation of the three curvatures in the forms and or, equivalently, By the Pythagorean theorem, the absolute, normal, and geodesic curvatures are related by the equation

The last equation is illustrated in the following diagram, which shows the unit vectors \(\mathbf{N}\), \(\mathbf{n}\), and \(\mathbf{P}\) in the plane orthogonal to the curve. The vectors \(\mathbf{N}\), \(\mathbf{n}\) are orthogonal to each other, while \(\mathbf{P}\) may be oriented in an arbitrary way with respect to them.

The normal curvature on the surface of a sphere is the same for all curves at all points. If the radius of the sphere is \(R\), then the common value of normal curvature with respect to the outward normal is \(-1/R\). To show this, recall that on the surface of a sphere the curvature tensor is related to the metric tensor by the identity Therefore, Finally, since \(T^{\alpha}\) are the components of a unit vector, we have as we set out to show.

Thus, on the surface of a sphere of radius \(R\), the equation of three curvatures reads and therefore the absolute curvature \(\sigma\) satisfies the identity If the curve is a circle of radius \(r\), then the above equation becomes which is consistent with our earlier analysis of geodesic curvature of a circle on the surface of a sphere.

Suppose that a surface is cut by a plane at a point \(A\) and consider a curve that arises at the intersection of the surface and the plane. Meusnier's theorem, named after the French mathematician Jean Baptiste Meusnier, states that the absolute curvature \(\sigma\) of the resulting curve at the point \(A\) depends only on the curvature tensor \(B_{\alpha\beta}\) and the orientation of the plane.

The analytical portion of Meusnier's theorem follows from the equations of the three curvatures Dotting both sides with the surface normal \(\mathbf{N}\) yields In other words, Note that the curve's unit tangent \(\mathbf{T}\) and its principal normal \(\mathbf{P}\) both lie in the cutting plane and therefore determine its orientation. Therefore, the right side in the above identity indeed depends only on the curvature tensor and the orientation of the surface. Also note that the right side is independent of the choice of the surface normal \(\mathbf{N}\) since both \(B_{\beta}^{\alpha}\) and \(\mathbf{N}\) change their sign when the choice is reversed.

If \(\gamma\) is the angle between the principal normal \(\mathbf{P}\) and the surface normal \(\mathbf{N}\), then the above formula reads This is the usual form in which Meusnier's theorem appears. In particular, if the cutting plane contains the surface normal \(N\) then the absolute curvature matches the normal curvature up to sign, i.e.

Meusnier's theorem has a very striking geometric interpretation for which we need to introduce the concept of an osculating circle for a planar curve. At a given point \(A\) on a planar curve, the osculating circle is a circle that passes through \(A\), and has the same tangent and curvature as the curve. Analytically speaking, the curve and the circle agree in the first and second derivatives. The radius of the circle, i.e. the reciprocal of the absolute curvature, is referred to as the radius of curvature for the curve at the point \(A\). Meanwhile, the center of the circle is referred to as the center of curvature.

Now, consider a family of cutting planes passing through the point \(A\) that share an axis containing the tangent vector \(\mathbf{T}\). All of the resulting curves pass through the point \(P\) and share the tangent \(\mathbf{T}\). Since the curves share a common tangent direction, they all have the same normal curvature \(B_{\beta}^{\alpha}T_{\alpha}T^{\beta}\). Let us choose the surface normal \(\mathbf{N}\) so that \(B_{\beta}^{\alpha}T_{\alpha }T^{\beta}\) is negative and denote \(-B_{\beta}^{\alpha}T_{\alpha}T^{\beta}\) by \(1/R\), i.e. Then the absolute curvatures of the resulting curves are given by where \(\gamma\) is, once again, the angle between the principal normal \(\mathbf{P}\) and the surface normal \(\mathbf{N}\).

Observe that these are the very absolute curvatures we would observe if our surface was a sphere of radius \(R\) and, correspondingly, the curves arising as the result of the planar cuts were circles, as in the following figure. The fact that the absolute curvatures of the circles satisfy the equation can be seen in the following diagram which shows that orthogonal cross-section of the sphere. Thus, returning to our actual surface, if each of the curves is replaced with its osculating circle the collection of the circles will form a sphere of radius \(\left( B_{\beta}^{\alpha}T_{\alpha}T^{\beta}\right) ^{-1}\).

Recall Weingarten's equation for the surface covariant derivative of the normal \(\mathbf{N}\). The component form of Weingarten's equation reads Let us derive the analogous equations for the geodesic normal \(\mathbf{n}\).

For reasons that will become apparent shortly, we will first present the component form of Weingarten's equation which reads Thus, its form is entirely analogous to its classical counterpart. Since the same is true for its derivation, we will leave it as an exercise.

The vector form of Weingarten's equation, on the other hand, is fundamentally different from its classical counterpart, owing to the lack of the metrinilic property of \(\nabla_{\Phi}\) with respect to the surface basis \(\mathbf{S} _{\alpha}\). Indeed, differentiating both sides of we find Recall that Thus, and, equivalently, The surprising aspect of the formula is that, unlike its component form, it is more complicated than its classical counterpart. Of course, this is an artifact of the curvature of the ambient surface which is captured by the term containing the curvature tensor.

Exercise 9.1From the equation show that as well as the equivalent forms and

Exercise 9.2Show that \(n^{\alpha}\) given by satisfies the equations \(n_{\alpha}S_{\Phi}^{\alpha}=0\) and \(n_{\alpha }n^{\alpha}=1\).

Exercise 9.3Using the approach introduced in Chapter 2 for the proof of the projection formula prove the geodesic projection formula and establish its component form

Exercise 9.4Prove the component form of the geodesic projection formula from the explicit expression for the components \(n^{\alpha}\) of the geodesic normal.

Exercise 9.5Derive the equation from by dotting both sides with \(\mathbf{S}^{\Delta}\) to obtain Then obtain the desired result, by contracting both sides with \(S_{\Delta }^{\beta}\) and applying the geodesic projection formula.

Exercise 9.6Derive Weingarten's equation for the geodesic normal, i.e.

Exercise 9.7Consider a curve on the surface of a sphere of radius \(R\) referred to spherical coordinates \(\theta,\varphi\) given by equations Show that its geodesic curvature is given by Show that in the special case of constant \(\theta\), i.e. \(\theta\left( \gamma\right) =\theta_{0}\), this formula agrees with the equation obtained earlier.