A Riemannian space is a coordinate space that is not connected to a Euclidean space or one whose connection to the Euclidean space from which it emerged is ignored. Riemannian spaces make it possible to use the ideas and the internally consistent analytical framework that we have developed without being beholden to a Euclidean geometric reality. The Riemannian framework frees us from adhering to the assumptions underpinning Euclidean geometry and, in fact, from our inability to exceed three dimensions. As such, Riemannian spaces are a powerful metaphorical extension of analytical Euclidean Geometry capable of describing realities that deviate from the Euclidean paradigm. This is a powerful generalization that is essential for us if we believe, as we have for over one hundred years, that the physical space of our existence is, in fact, non-Euclidean

At this point in our narrative, the concept of a Riemannian space seems all but inevitable. After all, with the exception of Chapter 18, we have been operating almost exclusively in the coordinate space for quite some time now, only occasionally checking in with the original Euclidean space. The issues that have preoccupied our attention, in particular the tensor property, required analyses that relied on the properties of the metric tensor $Z_{ij}$, but not its definition. It seems natural, then, to ask what would happen if $Z_{ij}$ was arbitrarily selected rather than calculated by pairwise inner products of the covariant basis vectors $\mathbf{Z}_{i}$? The short answer is that the overall framework remains intact and, in fact, becomes richer and more interesting.

We should note, however, that the apparent inevitability of Riemannian space is counter-historical. Indeed, it is the Riemannian ideas that led to the development of Tensor Calculus, the latter being first and foremost a tool for organizing those ideas into an analytical framework. Meanwhile, the fact that Riemannian spaces are being discussed in Chapter 20 is just another illustration that mathematical textbooks usually tell ideas in reverse-historical order.

Let us briefly recount our journey to Riemannian spaces. Our initial point of departure was the concept of a Euclidean space -- the geometric space of our everyday physical existence in which Euclid's axioms are found to have a reasonable degree of internal consistency and their consequences reasonably approximate reality as we perceive it. A Euclidean space is automatically endowed with the concepts of length and angles from which we construct the operation of the dot product of two geometric vectors $\mathbf{U}$ and $\mathbf{V}$, i.e.

The imposition of a coordinate system upon a Euclidean space gives rise to the coordinate space -- a region in $\mathbb{R} ^{1}$, $\mathbb{R} ^{2}$, or $\mathbb{R} ^{3}$ that represents the coordinates of the points in the Euclidean space and gives rise to a slew of analytical objects, such as the covariant and the contravariant bases $\mathbf{Z}_{i}$ and $\mathbf{Z}^{i}$, the covariant and the contravariant metric tensors $Z_{ij}$ and $Z^{ij}$, the volume element $\sqrt{Z}$, the Christoffel symbol $\Gamma_{jk}^{i}$, and the Levi-Civita symbols $\varepsilon_{ijk}$ and $\varepsilon^{ijk}$. These objects are said to be induced from the Euclidean space. The fact that these objects arose in the context of a Euclidean space means that their values are not arbitrary, but rather subject to numerous constraints. For example, the metric tensor $Z_{ij}$ is symmetric, i.e. and positive-definite. Consequently, the Christoffel symbol $\Gamma_{jk}^{i}$ is symmetric in its subscripts, i.e. And finally, the foremost of all constraints is the Riemann-Christoffel identity derived at the end of Chapter 15.

Over the course of our narrative, we gradually transitioned from working with vectors in the Euclidean space to working with their components in the corresponding coordinate space. In particular, once all vectors are converted into their components, we are able to leave all vector quantities, including the bases $\mathbf{Z}_{i}$ and $\mathbf{Z}^{i}$, completely out of the analysis and proceed only with objects that are available in the coordinate space. We have found that all analyses that can be performed geometrically in the Euclidean space can also be performed algebraically in the coordinate space. (In fact, the coordinate space offers far more robust analytical and computational tools than the original Euclidean space where we are constrained to use geometric methods, which are quite limited.) Thus, coordinate spaces have proven to be self-sufficient. However, since a coordinate space analysis exactly parallels what happens in the Euclidean space, its results are always consistent with Euclid's axioms and their consequences.

Our next important observation was the almost all-encompassing role played by the metric tensor $Z_{ij}$. Thanks to the formula the metric tensor $Z_{ij}$ is sufficient for the evaluation of dot products in terms of the components of vectors. This enables it to capture much of the geometry of the original Euclidean space. Furthermore, we discovered that the Christoffel symbol $\Gamma_{jk}^{i}$ can be expressed in terms of the covariant metric tensor $Z_{ij}$ and its derivatives, i.e. Thus, despite the fact that our original definition was vector-based, i.e. or, equivalently, and thus included references to objects not available in the coordinate space, the metric tensor can serve as an alternative starting point for the Christoffel symbol. Therefore, all operations in the coordinate space, including the covariant derivative can be built up strictly from the covariant metric tensor $Z_{ij}$ and its derivatives. On a practical level, this means that the connection between the original Euclidean space and the coordinate space can be completely severed once the metric tensor $Z_{ij}$ has been calculated by the formula The only time that a reference to the original Euclidean space may be needed is to gain the geometric interpretation of the final results obtained by the coordinate space analysis.

In summary, in order for a coordinate space to be truly self-sufficient, all that is needed is the metric tensor field $Z_{ij}\left( Z\right)$. This insight inevitably raises the intriguing possibility of starting out with a domain in $\mathbb{R} ^{n}$ and choosing $Z_{ij}\left( Z\right)$ arbitrarily (albeit subject to some desirable conditions, such as symmetry, positive definiteness, and spatial continuity) while preserving the overall analytical framework. In other words, the idea is to apply all of the machinery that we have developed so far to a "coordinate" space with a metric tensor field that is not necessarily induced from a Euclidean space.

Will this algebraic construct exhibit the same level of internal consistency as a Euclidean coordinate space or should we expect to run into insurmountable contradictions? On the one hand, it may be reasonable to hope for internal consistency since we are only changing the input while preserving the framework. On the other hand, the internal logical consistency of a Euclidean coordinate space is buttressed by the internal logical consistency of the Euclidean space itself. By assigning a metric tensor field that is not induced from a Euclidean space, we are no longer able to look to a Euclidean space for an absolute geometric interpretation of our results, a reference that has served as a reliable source of internal consistency.

Fortunately, even in the absence of a Euclidean space, the tensor framework remains. After all, the tensor property has to do with the transformation from one coordinate system to another and, therefore, the concept of a tensor is relative and does not require the absolute reference of a Euclidean space. The concept of a tensor survives virtually unchanged while the ultimate concept of an invariant preserves the meaning of having the same value in all coordinate systems but foregoes the greater implication of having a coordinate-free geometric interpretation. Nevertheless, this is sufficient for providing the new framework with internal logical consistency.

Meanwhile, it is clear that the results of a Riemannian analysis may deviate from the experiences of our everyday physical existence that is consistent with the axioms and the conclusions of Euclidean Geometry. For the most striking example, note there is no longer reason to expect that the Riemann-Christoffel identity will continue to hold as it did in any Euclidean coordinate space. Recall that this identity is an expression of the "straightness" of a Euclidean space and the consequent admissibility of an affine coordinate system. Thus, violation of the above identity for an arbitrarily selected metric tensor field $Z_{ij}$ signals that it does not correspond to any Euclidean coordinate space.

This feature of Riemannian space is, of course, an unequivocal positive. Whether the physical space of our everyday existence is accurately described by Euclidean Geometry has been the subject of continuous skepticism for over two thousand years. According to some models, most notably Einstein's Theory of Relativity, our space violates some of the assumptions underpinning Euclidean Geometry and thus some of its conclusions. Riemannian spaces therefore hold the potential for supporting more general theories of space.

A Riemannian space is a strikingly minimalist algebraic construct. By definition, a Riemannian space is a domain in $\mathbb{R} ^{n}$ along with a metric tensor field $Z_{ij}\left( Z\right)$. The integer $n$ is referred to as the dimension of the space and can have any positive value. We will treat $Z_{ij}$ as the covariant metric tensor, although the Riemannian version of the concept of a tensor, and therefore the meaning of the term covariant, are yet to be clarified. We will always require the metric tensor $Z_{ij}$ to be symmetric, i.e. We will also typically require positive definiteness, although some applications -- including General Relativity -- require non-positive definite metric tensors. For our present purposes we will also assume that $Z_{ij}\left( Z\right)$ is sufficiently differentiable.

A number of crucial definitions from Euclidean coordinate spaces remain completely intact in the new Riemannian context. The contravariant metric tensor $Z^{ij}$ is defined as the matrix inverse of the covariant metric tensor $Z_{ij}$, i.e. The practice of index juggling by contraction with the metric tensors $Z_{ij}$ and $Z^{ij}$ is completely unchanged. The Christoffel symbol $\Gamma_{i,jk}$ is defined by the equation while $\Gamma_{jk}^{i}$ is given by From this definition, known as the intrinsic definition, it follows that The definition of the covariant derivative $\nabla_{k}$ is completely unchanged, i.e. and all of its properties remain intact.

The Riemann-Christoffel tensor $R_{\cdot mij}^{k}$, given by now takes on an even greater importance since it no longer vanishes. In fact, it serves as an important characterization of the Riemannian space and will occupy a central place in our analysis below.

The volume element is $\sqrt{Z}$, where $Z$ is the determinant of the covariant metric tensor $Z_{ij}$. Finally, the Levi-Civita symbols $\varepsilon_{ijk}$ and $\varepsilon^{ijk}$ are given by Recall that the order of the Levi-Civita symbols matches the dimension of the space.

Let us now turn our attention to the differences between the ways in which Euclidean coordinate spaces and Riemannian spaces are constructed. We will find that while most of the identities are exactly the same, their interpretations are different: equations that once served as corollaries will now serve as definitions.

Let us begin by taking a second look at the symmetry requirement for the covariant metric tensor $Z_{ij}$. In a Euclidean coordinate space, the symmetry of $Z_{ij}$ is a corollary of its definition, i.e. $Z_{ij} =\mathbf{Z}_{i}\cdot\mathbf{Z}_{j}$. In a Riemannian space, it is part of the definition itself. The same can be said of the positive definiteness of $Z_{ij}$.

In a Riemannian space, a vector is any first-order system $U^{i}$. (Below, we will refine this definition by requiring $U^{i}$ to be a tensor.) This definition clearly illustrates the loss of the absolute reference that we enjoyed in the Euclidean context. Recall that in a Euclidean coordinate space, a tensor $U^{i}$ can be converted into the corresponding invariant geometric vector $\mathbf{U}$ by the contraction $U^{i}\mathbf{Z}_{i}$. This gives $U^{i}$ an instant absolute interpretation independent of the coordinate system. Such an interpretation is no longer available. In a Riemannian space, vectors behave in the same way as components of geometric vectors behave in a Euclidean coordinate space. However, beyond this algebraic similitude, the two concepts are distinct and it is essential to accept Riemannian vectors on their own terms, as it was essential to accept Euclidean vectors on their own terms in Chapter 2.

For the concepts of length, angle, and dot product, the logic of Euclidean coordinate spaces is completely reversed, `{a} la the Linear Algebra approach described in Section 2.7. For two vectors $U^{i}$ and $V^{i}$, the dot product is defined to be the combination Thus, this familiar contraction has changed its role from a corollary to a definition. Although it is called the dot product, there is no symbol for this operation in a Riemannian space that includes a dot. It is easy to show that the above definition satisfies the requisite properties of an inner product in the sense of Linear Algebra.

The length of a vector $U^{i}$ is the square root of the dot product with itself, i.e. The angle $\gamma$ between two vectors $U^{i}$ and $V^{i}$ is given by the equation (Note that, technically speaking, each dot product above should use its own set of dummy index names.) The fact that the absolute value of the quantity on the right is less than or equal to $1$ is an immediate consequence of the Cauchy-Schwarz inequality. Two vectors $U^{i}$ and $V^{i}$ are said to be orthogonal if their dot product vanishes, i.e. An ordered set of $n$ linearly independent vectors is said to be positively oriented, if the determinant of the matrix consisting of the $n$ vectors is positive.

With the help of index juggling, which, as we have already mentioned, remains completely intact, the above definitions can be expressed in a more concise form. The dot product of $U^{i}$ and $V^{i}$ is given by the length of a vector $U^{i}$ is and the angle $\gamma$ between two vectors $U^{i}$ and $V^{i}$ satisfies the equation (Once again, each dot product should have used its own set of dummy index names.) Vectors $U^{i}$ and $V^{i}$ are orthogonal if

A set of $n$ functions $Z^{i}\left( \gamma\right)$ of a single parameter $\gamma$ is referred to as a curve. The integral or, more explicitly, is, by definition, the length of the curve over the interval from $\gamma_{1}$ to $\gamma_{2}$.

Let us return to the Riemann-Christoffel tensor It was initially introduced in Chapter 15 in the context of Euclidean coordinate spaces. We were therefore able to conclude that it identically vanishes in all coordinate systems. Now that we are no longer in a Euclidean coordinate space, the Riemann-Christoffel tensor does not vanish and therefore takes on an even greater significance. Recall that the original purpose of the Riemann-Christoffel tensor was to express the commutator $\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i}$ in terms of the Christoffel symbols and we discovered that This remarkable formula was first given by Gregorio Ricci and Tullio Levi-Civita in their 1901 seminal paper titled M\'{ethodes de calcul differ'{e}ntiel absolu et leurs applications}.

In this Section, we will discuss the most fundamental properties of the Riemann-Christoffel tensor. Most of the details of the demonstrations will be left as exercises.

Let us focus on the version of the Riemann-Christoffel tensor with a lowered first index, i.e. By absorbing the metric tensor into the Christoffel symbol, we find that

The Riemann-Christoffel tensor $R_{lmij}$ has three fundamental symmetries. The first one is the skew-symmetry in the last two indices, i.e. which is nearly self-evident from the previous equation. It is not self-evident, but nevertheless true, that $R_{lmij}$ is also skew-symmetric in the first two indices, i.e. and symmetric with respect to switching the first two with the last two indices Furthermore, $R_{lmij}$ has an additional symmetry captured by the first Bianchi identity As a result of these symmetries, the number of the degrees of freedom in the Riemann-Christoffel tensor equals $n^{2}\left( n^{2}-1\right) /12$. In particular, in a two-dimensional Riemannian space, the Riemann-Christoffel tensor has a single degree of freedom, a fact that we will take full advantage of below and that will lead to the concept of Gaussian curvature. The proofs of the above symmetries as well as the statement regarding the degrees of freedom are left as exercises.

The Ricci curvature tensor is the second-order tensor $R_{mj}$ obtained from the Riemann-Christoffel tensor by contracting the first and the third indices, i.e. It is easy to show that the Ricci curvature tensor is symmetric, i.e. Its trace $R$, i.e. is known as the scalar curvature.

Recall that the Riemann-Christoffel tensor originally emerged during the analysis of the commutator in Chapter 15, where we established the following identity for a first-order variant $T^{k}$: This identity can be extended to variants of arbitrary order. In the general identity, there is a term on the right for each index of the variant. Thus, as we always do in situations like this, we will give the formula for a variant $T_{j}^{i}$ with a representative collection of indices. The identity reads Thus, from the point of view of indicial logistics, the Riemann-Christoffel symbol figures on the right are in a way similar to the Christoffel symbol in the definition of the covariant derivative, where the sign of the term and the interplay among indices depends on the flavor of the index.

The concept of the Gaussian curvature applies in two-dimensional Riemannian spaces. The skew-symmetric properties of the Riemann-Christoffel tensor $R_{lmij}$, i.e. reduce it to a single degree of freedom. Indeed, there are only four nonzero elements, i.e. where Therefore, $R_{lmij}$ can be captured by the identity where $e_{lm}$ and $e_{ij}$ are the permutation systems described in Chapter 16. (Note that we encountered similar identities rooted in the skew-symmetric property in Section 16.10.) The permutation systems $e_{lm}$ and $e_{ij}$ can be easily converted into the Levi-Civita symbols Thus, we have where $K=\alpha/Z$. The quantity $K$ is known as the Gaussian curvature and is, perhaps, the single most important pointwise characteristic of a two-dimensional Riemannian space.

Multiplying both sides of the equation by $\varepsilon^{lm}\varepsilon^{ij}$ yields an explicit expression for the Gaussian curvature $K$, i.e. It also proves that $K$ is an invariant.

The Ricci curvature tensor, defined by the identity is given by Therefore, the Gaussian curvature equals half the scalar curvature, i.e. Proving these identities is left as an exercise.

First, consider the Riemannian space that occupies all of $\mathbb{R} ^{3}$ with the metric tensor $Z_{ij}$ that corresponds to the matrix This space obviously corresponds to a three-dimensional Euclidean space referred to Cartesian coordinates. Such Riemannian spaces are said to be induced from a Euclidean space. Since we have encountered this space before, there is no need to recount the values of $Z^{ij}$, $\Gamma_{jk}^{i}$, and other objects.

A related example of a Riemannian space is $\mathbb{R} ^{4}$ with $Z_{ij}$ corresponding to On the one hand, it is abundantly clear that for a wide range of intents and purposes this Riemannian space is analogous to the one before it. On the other hand, by virtue of having a dimension greater than $3$, this Riemannian space does not correspond to any Euclidean coordinate space. This example illustrates one aspect of the generality of Riemannian spaces. Furthermore, this Riemannian space offers us an opportunity to construct an algebraic generalization of Euclidean spaces more closely resembling the geometric space. We will exploit this idea below in Section 20.7. We must acknowledge in advance that this is precisely the kind of association between Euclidean spaces and regular Cartesian grids that we discouraged so strenuously in the beginning of the book. However, such an association is perilous when it is the exclusive interpretation or even the default one and not when it is just one among many.

For a second example, consider the space $\mathbb{R} ^{3}$ with $Z_{ij}$ corresponding to Similarly to the previous example, this Riemannian space is related to cylindrical coordinates imposed upon a three-dimensional Euclidean space. We are already familiar with the values of the fundamental elements in this space and will therefore not list them here. Note however, that this space has a singularity at all points corresponding to $Z^{1}=0$ since at those points, the above matrix is not invertible and therefore the contravariant metric tensor $Z^{ij}$ does not exist.

For a third example, consider $\mathbb{R} ^{2}$ with $Z_{ij}$ corresponding to where $R$ is a constant. Although the general form of $Z_{ij}$ is reminiscent of spherical coordinates, we have not encountered this space before and will therefore summarize the values of the fundamental object. The details of the calculations are left as an exercise.

The volume element $\sqrt{Z}$ is given by while the contravariant metric tensor $Z^{ij}$ corresponds to The nonzero elements of the Christoffel symbol $\Gamma_{jk}^{i}$ are Finally, the Riemann-Christoffel tensor $R_{kmij}$ is given by Thus, the Gaussian curvature $K$ is given by

The fact that the Riemann-Christoffel tensor $R_{kmij}$ does not vanish indicates that this Riemannian space does not correspond to any Euclidean coordinate space. Interestingly, it does correspond to a two-dimensional curved surface embedded in a three-dimensional Euclidean space. Of course, embedded surfaces are not discussed in this book and we are therefore not in a position to give a detailed description. However, we ought to mention the most basic facts because it is a very common way in which non-Euclidean Riemannian spaces can arise.

Consider a sphere of radius $R$ in a three-dimensional Euclidean space. Much like the points in the surrounding space, the points on the sphere can be enumerated by a pair of numbers. In other words, the sphere can be referred to a coordinate system. We will denote the coordinates by $S^{1}$ and $S^{2}$ or, collectively, $S^{\alpha}$ where a different alphabet is used for indices because the dimension of the sphere is different from the dimension of the surrounding space. The surface covariant basis $\mathbf{S}_{\alpha}$ is defined by the equation and the rest of the framework is constructed by following the Euclidean blueprint. For example, the covariant metric tensor $S_{\alpha\beta}$ is defined by and so on.

Now, when the sphere is referred to spherical coordinates $S^{1},S^{2} =\theta,\varphi$ illustrated in the figure the resulting metric tensor $S_{\alpha\beta}$ corresponds to which is precisely the metric tensor in the present example. Therefore, this Riemannian space may be said to be induced from a surface embedded in a Euclidean space. This example shows that some surfaces embedded in Euclidean spaces are themselves examples of non-Euclidean spaces. Note that the Gaussian curvature for embedded surfaces has a beautiful geometric interpretation that will be discussed in a future book.

Before we can discuss the concept of a tensor, we must first talk about coordinate changes. Heretofore, our analysis of coordinate changes has been based on the equations of coordinate transformation that related two alternative coordinate systems: the "original" unprimed coordinates $Z^{i}$ and the "new" primed coordinates $Z^{i^{\prime}}$. However, it is important to understand the crucial role that the Euclidean space played in establishing these equations. Given two alternative coordinate systems $Z^{i}$ and $Z^{i^{\prime}}$, consider a specific set of unprimed coordinates, such as $\left( Z^{1},Z^{2},Z^{3}\right) =\left( 2,1,5\right)$. How does one determine the values of the primed coordinates $Z^{i^{\prime} }$ that correspond to the above values of the unprimed coordinates? This is done in two steps. The first step is to identify the physical point $P$ in the Euclidean space corresponding to unprimed coordinates $\left( 2,1,5\right)$. The second step is to observe the primed coordinates corresponding to the same point $P$, e.g. $\left( Z^{1^{\prime}},Z^{2^{\prime}},Z^{3^{\prime} }\right) =\left( 3,-2,8\right)$. By this mechanism, the function $Z^{i^{\prime}}\left( Z\right)$ maps $\left( 2,1,5\right)$ to $\left( 3,-2,8\right)$, while $Z^{i}\left( Z^{\prime}\right)$ maps $\left( 3,-2,8\right)$ back to $\left( 2,1,5\right)$.

With the Euclidean space no longer in the picture, we will take the equation as the definition, rather than a description, of the coordinate change. The range of the function $Z^{i^{\prime}}\left( Z\right)$ defines the domain of the "new" primed Riemannian space with coordinates $Z^{i^{\prime}}$. The inverse mapping function is denoted by $Z^{i}\left( Z^{\prime}\right)$, i.e. Naturally, the definitions of the Jacobians $J_{i}^{i^{\prime}}$ and $J_{i^{\prime}}^{i}$ remain the same as before:

The definition of a tensor under such changes of coordinates will remain the same as before, i.e. $T_{j}^{i}$ is a tensor with a representative collection of indices if $T_{j^{\prime}}^{i^{\prime}}$ and $T_{j}^{i}$ are related by the identity However, the application of this condition to some objects is not the same as before as the following example will illustrate.

Consider a Cartesian Riemannian space, i.e. $\mathbb{R} ^{n}$ with the metric tensor field $Z_{ij}$ that corresponds to the matrix Consider the following equations of coordinate transformation Of course, we recognize these equations as a transformation from Cartesian coordinates to cylindrical. For example, the point $\left( 1,1,3\right)$ maps to the point $\left( \sqrt{2},\pi/4,3\right)$. Thus, the primed Riemannian space is associated with the domain $\left[ 0,\infty\right) \times\left[ 0,2\pi\right) \times\left( -\infty,\infty\right)$. The inverse equations of coordinate transformation are, of course,

The primed Riemannian space is not yet complete since we have not specified the metric tensor field $Z_{i^{\prime}j^{\prime}}\left( Z^{\prime}\right)$. Since our goal is to preserve the tensor framework, $Z_{i^{\prime}j^{\prime} }\left( Z^{\prime}\right)$ cannot be assigned arbitrarily, nor can it be assigned by mapping the unprimed metric tensor $Z_{ij}$ to the primed space (which, in this example, would mean that $Z_{i^{\prime}j^{\prime}}$ also corresponds to the $3\times3$ identity matrix). The only logical thing to do is to construct the metric tensor $Z_{i^{\prime}j^{\prime}}$ in the primed coordinate space according to the equation Notice that this equation was not given the number (13.59) of the same equation in Chapter 13 to highlight the fact that it now serves as a definition rather than a corollary. Also note that in this approach, the coordinates $Z^{i}$ are truly primary while all other coordinate systems are secondary. This loss of parity among coordinate systems is caused by the absence of an independent Euclidean space acting as an a priori absolute reference.

To calculate the elements of $Z_{i^{\prime}j^{\prime}}$, note that Therefore, $Z_{i^{\prime}j^{\prime}}$ corresponds to matrix product which evaluates to the familiar matrix

In summary, we define the metric tensor $Z_{i^{\prime}j^{\prime}}$ so as to make $Z_{ij}$ transform according to the tensor rule. Similarly, we will augment the notion of a vector $U^{i}$ by stipulating that it transforms according to the tensor rule In other words, a Riemannian vector is, by definition, a first-order tensor. In general, for every object defined arbitrarily in the unprimed Riemannian space, we must specify the precise rule by which the object transforms under a change of coordinates to the primed space.

For all other objects, the question of transformation under a change of coordinates can be subjected to the same analysis as before and will reach the same conclusions. This applies to such objects as the contravariant metric tensor $Z^{ij}$, the volume element $\sqrt{Z}$, the Christoffel symbol $\Gamma_{jk}^{i}$, the Riemann-Christoffel tensor $R_{\cdot mij}^{k}$, and the Levi-Civita symbols $\varepsilon_{ijk}$ and $\varepsilon^{ijk}$. Once the metric tensor $Z_{i^{\prime}j^{\prime}}$ is constructed in the primed Riemannian space, the objects $\sqrt{Z^{\prime}}$, $\Gamma_{j^{\prime }k^{\prime}}^{i^{\prime}}$, $R_{\cdot m^{\prime}i^{\prime}j^{\prime} }^{k^{\prime}}$, $\varepsilon_{i^{\prime}j^{\prime}k^{\prime}}$, and $\varepsilon^{i^{\prime}j^{\prime}k^{\prime}}$ can be constructed from it. Thus, we can study the rules by which they transform between the two spaces and, because they have the same definitions as they did in Euclidean coordinate spaces, we will, of course, discover that they transform by the exact same rules.

Finally, it must be noted that the concept of a tensor is stronger and more absolute in a Euclidean space than in a Riemannian space. In a Euclidean space, an invariant constructed from tensors represents an object with a clear geometric meaning. Thus, the tensor framework connects the coordinate space to an independent physical reality which serves as a guarantor of the framework's meaningfulness. In a Riemannian space, there is no such independent physical reality that can be used as an absolute reference. Thus, the tensor framework serves as a tool for transferring the principles developed in Euclidean spaces to a new algebraic setting. It brings with it a certain degree in internal consistency. For example, the tensor property continues to be reflexive, symmetric, and transitive as described in Section 14.12. On balance, however, the tensor framework in the context of a Riemannian space should be seen as a metaphorical extension of Euclidean ideas.

A Riemannian space is, first and foremost, an algebraic structure. In a way, its very purpose is to provide a framework where analysis can be performed without a need for geometric intuition. It is somewhat ironic, then, that, being a subset of $\mathbb{R} ^{n}$, a Riemannian space can be visualized as a Cartesian coordinate grid in a Euclidean space. And not only that, this geometric view of a Riemannian space is actually beneficial.

For example, consider the Riemannian space induced from a Euclidean plane referred to polar coordinates $r,\theta$. In the spirit of Riemannian spaces, we will refer to the coordinates as $Z^{1}$ and $Z^{2}$ instead of $r$ and $\theta$. The induced Riemannian space occupies the domain $\left[ 0,\infty\right) \times\left[ 0,2\pi\right)$. The metric tensor $Z_{ij}\left( Z\right)$ corresponds to the matrix

When asked to visualize the domain $\left[ 0,\infty\right) \times\left[ 0,2\pi\right)$, all of us, without exception, visualize a semi-infinite rectangular strip of height $2\pi$. Furthermore, the points corresponding to integer (or any regularly spaced) values of $Z^{1}$ and $Z^{2}$ form a regular grid. The result is nothing but a Euclidean space referred to Cartesian coordinates. We will refer to this space as the arithmetic space associated with the Riemannian space. Of course, any Riemannian space, not just induced ones, may be visualized as an arithmetic space. The metric tensor, an essential part of the Riemannian space, is not reflected in this picture and thus remains behind the scenes. Surprisingly, the geometry that takes place in the arithmetic space will play an important role in many of our future analyses. For example, the divergence theorem will be proven by first demonstrating it in the arithmetic space.

We must point out the obvious that for induced Riemannian spaces, the arithmetic geometry may look quite different from the original geometry from which the Riemannian space had arisen. For example, consider a curve in a Euclidean plane given by the following equations in polar coordinates Of course, we recognize this shape as a spiral On the other hand, if we consider the curve in the arithmetic space given by the same equations, we will observe a straight line: Thus, the actual curve and its arithmetic manifestation have two completely different shapes. In other words, the arithmetic space corresponding to an induced Riemannian space is a gross distortion of the associated Euclidean space.

A more vivid example of the same effect is the familiar distortion that occurs when the spherical Earth is represented on a flat map. The distortion is especially apparent near the poles where the volume element $\sqrt{Z} =R^{2}\sin\theta$ is particularly small. The degree to which the shape of a great circle can change is also a striking illustration of the deformation. The great circle in the figure above is the terminator, i.e. the line that separates day and night.

Let us be reminded however that, despite the discrepancy between the Euclidean and the corresponding arithmetic spaces, the availability of the metric tensor $Z_{ij}$ assures that the arithmetic space retains all of the necessary information to perform Euclidean analysis. For example, suppose that the equation described the trajectory of a material particle. Given these equations of motion we cannot easily imagine the actual Euclidean trajectory of the material particle. Nevertheless, we can calculate the components of its velocity and acceleration and calculate their magnitudes. Specifically, the components of the velocity are given by the equation while the components of the acceleration are given by Note that the availability of the Christoffel symbol is crucial for the latter calculation. With the help of the metric tensor, the magnitude of the velocity is given by while that of the acceleration is given by We can also calculate the total distance $L$ traveled between the times $t_{1}$ and $t_{2}$ according to the equation In summary, the fact that the arithmetic space is severed from the Euclidean space does not in any way preclude us from doing virtually any kind of analysis short of visualizing the actual shapes.

Finally, going slightly beyond our present scope, note that in terms of the equations of motion $Z^{i}\left( t\right)$, the components $A^{i}$ of acceleration are given by Thus, if the particle is moving uniformly in the Euclidean space and therefore its acceleration is zero, then the equations of motion are characterized by the identity known as the geodesic equation. In a Euclidean space, the shortest curve connecting two points is a straight line and, thus, the geodesic equation tells us whether $Z^{i}\left( t\right)$ represents uniform motion along a straight line.

In a general Riemannian space, the length $L$ of a section of a curve given by $Z^{i}\left( t\right)$ is defined by the integral It can be shown that the geodesic equation represents a form of the Euler-Lagrange equation for the above integral and thus characterizes the shortest path, again in the Riemannian sense, between two points. The discussion of the geodesic equation is beyond the scope of this book because it is best discussed in the context of the Calculus of Moving Surfaces where it can be given a more general treatment.

At the very outset of our narrative, we defined a Euclidean space as the physical space of our everyday existence in which the axioms of Euclidean geometry are valid. In particular, we have relied heavily on two unique characteristics of Euclidean spaces. First, Euclidean spaces can accommodate straight lines in any direction. This enabled us to introduce, and work freely with, geometric vectors. Secondly, Euclidean spaces admit Cartesian coordinate systems characterized by regular orthogonal coordinate grids. As a result, the metric tensor corresponds to the identity matrix at all points.

Between these two advantages, the former outweighs the latter. The methods of Tensor Calculus almost completely mitigate the effects of complicated coordinate systems. In fact, this subject's commitment to avoiding the use of special features of coordinate systems is one of the keys to its success. On the other hand, the use of geometric vectors was instrumental in stimulating our geometric intuition and helping us organize information in a very appealing way. Compared to the collection of coordinates $U^{\alpha}$, the geometric vector $\mathbf{U}$ is a tangible object that connects our analytical investigations to our geometric intuition. Furthermore, compared to $U^{\alpha}$, $\mathbf{U}$ is a variant of lower order and is therefore simpler. Finally, geometric vectors help guide our analytical intuition in manipulating coordinate expressions as our mind becomes trained to look for the geometric-vector interpretation.

Unfortunately, Euclidean spaces, according to our geometric definition, are limited to three or fewer dimensions. In higher dimensions, the only construct available to us is the Riemannian space. However, in a Riemannian space, i.e. a subset of $\mathbb{R} ^{n}$ paired with an arbitrary metric tensor field, there is no such thing as a geometric vector. In the absence of geometric vectors, a first-order tensor $U^{\alpha}$ is condemned to variance for a greater part of our analysis. We can determine its invariant dot product with another vector $V^{\beta}$ as $S_{\alpha\beta}U^{\alpha}V^{\beta}$ and its invariant length as the square root of the dot product with itself, but it otherwise remains in its intermediate variant state devoid of an immediate geometric interpretation.

However, having learned the great utility of boldface symbols in organizing our ideas in the Euclidean space, we would like to extend it to Riemannian spaces. This can be accomplished by introducing a construct referred to as the arithmetic Euclidean space in the following way. An arithmetic Euclidean space is a Riemannian space that occupies the entirety of $\mathbb{R} ^{n}$ where the metric tensor $Z_{ij}$ corresponds to the $n\times n$ identity matrix. At all points, the elements of the covariant basis, denoted by $\mathbf{Z}_{i}$, correspond to the $i^{\text{th}}$ column of the identity matrix. For example, in $\mathbb{R} ^{4}$, Then, for a tensor $U^{i}$ of order one, the invariant $\mathbf{U}$ is defined by

As an illustration, we will extend the Frenet formulas to higher dimensions. Recall that the Frenet equations read where the subscript $s$ denotes differentiation with respect to arc length $s$.

Part of the what makes the Frenet equations so elegant is the skew-symmetric property of the matrix. It turns out that this feature persists to higher dimensions.

Consider a curve embedded in an $n$-dimensional arithmetic Euclidean space. Suppose that the curve is referred to its arc length $s$ and is specified by the equations Let $\mathbf{T}_{1}$ be the unit tangent to the curve. Then This definition enables us to avoid introducing the position vector $\mathbf{R}$, which can certainly be done but is not necessary. Subsequently, however, we will not refer to the components of $\mathbf{T}_{1}$ or any of the other vector quantities going forward.

The identity above defines the initial vector $\mathbf{T}_{1}$ from the local frame. Subsequently, the unit vector $\mathbf{T}_{m+1}$ is obtained from $\mathbf{T}_{m}^{\prime}\left( s\right) \$by applying the Gram-Schmidt algorithm in order to make it orthogonal to each of the preceding vectors, and subsequently factoring out a positive scalar to make it unit length. Finally, the normality of $\mathbf{T}_{n}$ is assured in a slightly different way, that you may already anticipate.

Let us investigate the construction of $\mathbf{T}_{2}$ from $\mathbf{T} _{1}^{\prime}\left( s\right)$. Since $\mathbf{T}_{1}\left( s\right) \cdot\mathbf{T}_{1}\left( s\right) =1$, differentiating both sides with respect to $s$ yields $2\mathbf{T}_{1}\left( s\right) \cdot\mathbf{T} _{1}^{\prime}\left( s\right) =0$ from which it follows that $\mathbf{T} _{1}^{\prime}\left( s\right)$ is orthogonal to $\mathbf{T}_{1}$. Let $\kappa_{1}$ be the length of $\mathbf{T}_{1}^{\prime}\left( s\right)$ and let $\mathbf{T}_{2}$ be the unit vector that points in the same direction as $\mathbf{T}_{1}^{\prime}$, i.e.

Let us now move on to the next step and analyze$\ \mathbf{T}_{2}^{\prime}$ which, by a previously used argument, is orthogonal to $\mathbf{T}_{2}$, i.e. Differentiating this identity with respect to $s$ yields since $\mathbf{T}_{1}^{\prime}=\kappa_{1}\mathbf{T}_{2}$ and therefore $\mathbf{T}_{2}\cdot\mathbf{T}_{1}^{\prime}=\kappa_{1}\mathbf{T}_{2} \cdot\mathbf{T}_{2}=\kappa_{1}$, we find Thus, the vector is orthogonal to both $\mathbf{T}_{1}$ and $\mathbf{T}_{2}$ and can therefore be used to define the unit vector $\mathbf{T}_{3}$ and the corresponding absolute curvature $\kappa_{2}$ by the equation This identity can also be written in the form that will serve as the second Frenet formula.

We now embark on the inductive step of the procedure. The vector $\mathbf{T}_{m}$ is determined by the condition that it is unit length, i.e. and is orthogonal to all preceding vectors $\mathbf{T}_{k}$, i.e. Furthermore, we presume that for all $k$ between $1$ and $m-1$, we have the Frenet formulas