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m The Riemannian and on a s c i t a m e h t a Niles G. Johnson, 2003 David T. Guarrera, Northwestern University Homer F. Wolfe, New College of Florida Advisors: DaGang Yang and Morris Kalka (Tulane University) Department of

The computation of terms in a Taylor series illustrates an interplay between key geometric notions.

he main goal and result of our of flat Euclidean affixed at each point nian” means that the different inner prod- T work this summer was a com- on the manifold (see Figure 1(b)). This is a ucts change smoothly. With the metric in putation exhibiting the relation- generalization of the lines (m = 1) the we produce a notion of ship between the curvature tensor and the studied in every calculus course. and distance on the manifold itself. Riemannian metric on a smooth Rieman- To say that a manifold is Riemannian is The 2- (a mathematical model for the nian manifold. The most interesting and to say that it has a Riemannian metric, G = of a ball) is a . m exciting aspects of our work, however, were (gij) i,j = 1, which is an inner product on each When you say that the “straight lines” are the methods employed in the calculation. tangent space— it defines the measure of great circles and that each has length 2πr We will walk through an overview of these and distances in the tangent space. (where r is the radius of the sphere), you methods but leave most of the technical For vectors x, y in some tangent space, are recognizing the notion of angle and details in another paper. As we go, the goal G(x, y) = xGy, using matrix multiplication distance which is induced by the standard for which we worked is simple and should with x written as a row and y written as inner product on the tangent spaces. The be kept in mind. Differentiable functions a column. If p is the point whose tangent 2-sphere is in fact a smooth Riemannian can be written (in various ways) as infinite space we are considering, we may emphasize manifold; smooth means intuitively the

polynomials, and one of the standard ways this by writing the components of G as gij|p same thing as its English homonym, and

is called the Taylor series for the function. instead of just gij. This inner product need mathematically it means that everything in We looked at the Taylor series for a Ri- not be the same for each tangent space sight is differentiable (even the , emannian metric and showed that the terms (remember there is one for every point on and the derivatives of the derivatives, and in this polynomial are equal to expressions the manifold), but the adjective “Rieman- the . . .). involving only the curvature tensor and its covariant derivatives. ��� GROUNDWORK � The setting for our investigation is a � geometric object referred to above as a � smooth Riemannian manifold. There are complicated technical definitions of mani- folds, but the idea they describe is a place in which small regions look like (are closely approximated by) small regions in Euclid- ��� ean space of a certain , m, where m can be 1 (the real line, R), 2 (the plane, R2), 3 (R3), etc. The surface of a ball is a good example (for m = 2), because it shows how small pieces of a manifold could look like small pieces of a plane, while the whole manifold might not look like a plane (see Figure 1. A sphere is a good example of a smooth Riemannian manifold. a. A small Figure 1(a)). For each point on a manifold region of the sphere actually looks like a plane; b. At each point on a manifold, here we consider a tangent space, which is a copy our sphere, there is a tangent space.

30 2 Issue 1 Fall 2003 jur.rochester.edu 31 Taking derivatives on a manifold is The result we obtained expresses this tricky business, especially differentiating Taylor series in terms of the curvature tensor a tensor. We use two kinds of derivatives, and covariant derivatives of the curvature The curvature tensor and its covariant the partial , ∂, and the covariant tensor (covariant derivatives can be thought derivatives are expressed as derivative, , both of which are directional of as partial derivatives for a tensor). The derivatives. To describe how components general line of reasoning we follow is: of G change along a certain direction (say k x ) we use ∂k. One of the important facts 1. Use the 'right' coordinate system and about differentiation in is study G and R in that coordinate sys- that partial derivatives of a smooth vector tem. (4)

field commute; ∂k ∂l = ∂l ∂k for all l and k for each i, j, k1, …, kn. For n = 2 and n = 3, between 1 and m. On a general manifold, 2. Differentiate something called the we know the relationships more explicitly this is no longer true. The curvature tensor Gauss lemma to obtain a system of (the `polynomial in lower order partials' is a function, R, which describes the dif- equations involving appropriate par- turns out to be zero in each case).     ference between k l and l k; that is, tial derivatives of the gij. the degree to which the manifold deviates from flat Euclidean space (how `curved’ it 3. Examine the summation is). Whereas the metric has m2 components, the curvature tensor has m4 components, (5)

denoted Rikjl for some i, j, k, l, = 1, 2, …. Just Using the expression for covariant de- as with the components of the metric, each rivatives of R in terms of components

component of the curvature tensor is a real gij (and their partial derivatives), find

number (which may be different for differ- the highest-order derivatives of the gij (6) ent points on the manifold). For technical appearing in the summation above. reasons, we have to use the covariant de- THE GAUSS LEMMA rivative (instead of the partial derivative) to 4. Use relations derived from the Gauss The equalities above show that one determine how the components of R change lemma to reduce the terrible linear cannot simply substitute covariant deriva-

as we move across the manifold along some combination of highest-order deriva- tives of the R into the Taylor series for gij. line. All we will need to know here is that tives to a single highest-order deriva- Instead, we use the Gauss lemma to express a covariant derivative in the direction of xk tive. summations of the form found in the Taylor  is denoted k. series (equation 2) as summations of covari- Inductively, the lower order derivatives ant derivatives of R. AME LAN G P of the gij can be written in terms of lower Geometrically, the Gauss lemma says

The curvature tensor is derived from the order covariant derivatives of the Rikjl, and that straight lines emanating from a point metric, and the net result of our work is a subtraction then gives the result. on a manifold are perpendicular to loci of description of the opposite result— namely Explicitly, we will compute, for each n, constant distance from the point. Algebra- that the metric can be described in terms ically, this can be expressed as of the curvature tensor. We achieve this by considering the Taylor series for the met- (7) ric, which is a natural generalization of the Taylor series studied during the first couple DERIVATIVES OF THE GAUSS LEMMA of semesters of calculus. One of the most (3) The beauty of equation 7 is that it easily th interesting Taylor series is the expression for a particular constant Cn. allows us to derive dependencies of the n

for the exponential function. If t is a real order partials of gij at a base point p on the number and e = 2.718… denotes the base COORDINATES AND THE EXPRESSION FOR R manifold; this base point is the origin of our of the natural , then IN TERMS OF G coordinate system. Differentiating equation

In linear algebra one studies various 7 by ∂/∂xi, we get (1) choices of bases for vector spaces, and an im- For a manifold, the same idea is general- portant first step in our work was to choose ized slightly to higher by taking a nonstandard basis more naturally suited sums over different combinations of com- to express the geometric relationships un- Evaluating at the origin (base point), ponents. If p is a point on the manifold der consideration. We referred to this new we get and q is a point nearby whose coordinates basis as the orthonormal frame (sometimes 1 in Euclidean space with respect to p are x , called local canonical coordinates). In this gij(0) = δki (8) x2,…,xm then in reasonable circumstances orthonormal frame, we have some identities o h n s o n for G automatically; they are a result that is not terribly impressive, since it is a property of coordinates that j (2) we have already derived. Let us continue the

32 Volume 2 Issue 1 Fall 2003 jur.rochester.edu 33 m process by differentiating again, this time Substituting, the above simplifies to by ∂/∂ l. We get (i,l | s,k) + (l,s | i,k) + (s,i | l,k) = 0 x (16)

We can find the number of s distinct c i t a m e h t a Gauss equations that must exist for each Again, we evaluate at the base point to get order n. There is one Gauss equation for (20) each index that we choose to be fixed in our (9) tuple notation. We simply cycle through the We now combine terms in equation (17) We repeat the same process for the next other indices to get the n+1 tuples for an and examine the coefficient ofx kn…xk1 for a

order to get equation. Since there are n+2 indices that particular choice of indices k1,…, kn. It is we may choose to be fixed, there are n+2 (10) independent Gauss equations. (21)

We find that if we wish to obtain equa- Since, for any order, there are dis- with the summation over all σ ∈ Sn (that tions for the nth order partials of g at the tinct tuples (from the n+2 indices, we must is, all of the n! distinct arrangements of the

base point, we must differentiate the Gauss choose the n by which we are partially dif- indices k1, …, kn). Substituting from equa- Lemma n+1 times and then evaluate it at the ferentiating ), the number of dependent tion (4), this coefficient becomes origin. We introduce the useful notation partials is

In our notation, equations (9) and (10) Therefore, for any order, the Gauss Lemma become will allow us to find n+2 linear equations involving the partial derivatives of g at (i | l,k) + (l | i,k) = 0 (11) the origin. These equations can be used to eliminate partials and simplify (22) and calculations involving the partial deriva- tives of g. where hatted indices are omitted. (i,l | s,k) + (l,s | i,k) + (s,i | l,k) = 0 Permuting indices, we find systems of (12) equations. Among the nth-order partial APPLYING THE GAUSS LEMMA We have rearranged indices when permis- derivatives of g at p, this process produces a We use the nth-order Gauss lemma to th sible (using the facts that gij = gji and that system of n+2 simple equations in condense the n order partials of equation

partial differentials commute in Euclidean variables (derivatives of g). (22) to a single multiple of (kn,…, k1|i,j). space) in order to make the pattern appar- The second and third terms of equation (22) ent. In order to generate an nth order Gauss EXAMINING THE SUMMATION can be dealt with directly by two of the nth equation, pick n+2 indices. Fix one of these For each n, we determine the coefficient -order Gauss Lemma equations: indices; this letter will always index an entry of xkn … xk1 in the summation of g; that is, this entry will always stay on the right side of the horizontal bar in our (17) (23)

tuple notation. Keep adding successive lists, for a particular choice of k1,…,kn. and cyclically permuting the other n+1 indices. For n = 2, we use equation (5) and find For example, a third order Gauss equation the coefficient ofx kxl in to be would be (24) Simplification of the fourth term of (a,b,c | d,e) + (b,c,d | a,e) + (c,d,a | b,e) + equation (22) is more involved, but none- (d,a,b | c,e) = 0 (13) theless straightforward. The reader can check that the sum of all tuples is zero. Or rather which simplifies to Subtracting equations (23) and (24) from this sum shows that the sum of all terms that have both i and j to the left of “|” is equal (14) to the term having neither i nor j to the left An nth order Gauss equation is (18) of “|”. That is, Likewise, for n = 3, we use equation (6). The coeffecient of xk1xk2xk3 is

(15) (19) (25) SYSTEMS OF GAUSS EQUATIONS Equation (12) is not the only second order Gauss equation. We can do a simple permutation of indices to get

32 Volume 2 Issue 1 Fall 2003 jur.rochester.edu 33 All together this shows that equation Our work is a small part of a much the metric. Because the metric on a general (22) is equal to larger effort to understand . In manifold can be quite complicated, it is many ways manifolds are a natural gen- often difficult to compute this kind of in- eralization of Euclidean space and as such tegral. By considering the natural they are frequently the context for various of the particular manifold under discussion, (26) scientific studies. General manifolds are the curvature tensor may be more accessible which is to say that much more complicated than Euclidean than the metric. Expanding components of space (for Euclidean space, the curvature the metric in terms of the curvature tensor tensor is zero everywhere, as are all of its and its covariant derivatives allows one to derivatives), and mathematicians working use knowledge of the curvature tensor to in try to understand study these integrals. The moral of many how the various properties of a manifold are mathematical stories (like this one) is that By induction, this leads immediately to related. It is known that the curvature tensor much can be gained by looking at one thing equation (3), with determines all of the geometry on a mani- through its relationship to another. At its fold— including the metric— but it is not heart, our result is one such look at the fully understood how the curvature tensor curvature tensor. determines these things. Indeed, although Niles graduated from the University of Rochester in 2003 AFTERWORD our computation is more explicit than some with a B.A. and an M.A. in Mathematics. Homer, Dave, After surveying some of the explicit re- other results, it still does not give a closed and Niles completed their work on the Taylor expan- sion of a Riemannian Metric during the summer of sults, one may begin to wonder if research form for this determination. 2002 under the guidance of Professor DaGang Yang in mathematics is just endless technical These somewhat incomplete results do, at Tulane University. Their research was supported computation. This could not be more however, have an application. As one brief by an REU grant from the National Science Founda- wrong. The purpose of our research was example, we mention the use of manifolds tion. Niles is continuing his studies in a math Ph.D. program while his collaborators pursue careers in physics. to clarify of the relationship between two in the study of physics, especially in an qualities a manifold has— its metric and called string theory. In this area (as in other its curvature tensor. The computation is of physics, chemistry, etc.) scientists important only because it (partially) reveals study functions defined on a manifold and this relationship. try to integrate the functions with respect to o h n s o n j

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