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The Riemannian Metric and Curvature Tensor on a Manifold m The Riemannian Metric and Curvature Tensor on a Manifold a t h e m a t i c s 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 Mathematics 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 space 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 tangent lines (m = 1) the tangent space we produce a notion of ship between the curvature tensor and the studied in every calculus course. angle and distance on the manifold itself. Riemannian metric on a smooth Rieman- To say that a manifold is Riemannian is The 2-sphere (a mathematical model for the nian manifold. The most interesting and to say that it has a Riemannian metric, G = surface of a ball) is a Riemannian manifold. 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 angles 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 derivatives, 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 dimension, 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 Volume 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 derivative, ∂, 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 Euclidean space 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 logarithm, 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 dimensions 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 normal 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 distinct a t h e m a t i c s Gauss equations that must exist for each Again, we evaluate at the base point to get order n.
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