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Riemannian geometry a beginners guide - Morgan F.

Morgan F. Riemannian geometry a beginners guide - Jones and Bartlett, 1993. - 121 p.
ISBN 0-86720-242-4
Download (direct link): riemanniangeometry1993.djvu
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Thus the Ricci curvature has an interpretation as an average of sectional curvatures.
The scalar curvature R is defined as the trace of the Ricci curvature:
R = HRu. (10)
Hence for any orthonormal basis vt,... ,vm for TPS,
R = 2 1 K(v, = ^ [ K(P) (11)
1 vol Sr J
where 2P is the set of all 2-planes in TPS. Thus the scalar curvature is proportional to the average of all sectional curvatures at a point.
Remark. Historically Ric used to have the opposite sign. Some texts give the Riemannian curvature tensor Rijki the opposite sign.
5.3. The covariant derivative. Let S be a C2 m-dimensional surface in R". If / is a differentiable function on S, then the derivative Vu is a tangent vectorfield. But if f is a vectorfield (or a field of matrices or tensors), pointwise in TPS, then the derivative generally will have components normal to S. The projection into TPS is called the covariant derivative. See Figures 5.1 and 5.2. (The name comes from
certain nice transformation properties in a more general setting; see
Chapter 6.)
In local coordinates Ux,.. ., um in which g = I to first order at p, the coordinates of the covariant derivative of f at p are given by the several partial derivatives. For example, the coordinates /,;y of the covariant derivative of a vectorfield with coordinates /, are given by
Figure 5.1. (a) A vectorfield f on the circle, (b) its
derivative, and (с) its covariant derivative, 0.
Figure 5.2. (a) A vectorfield f on the circle, (b) its
derivative, and (c) its covariant derivative.
5.1. This problem studies the curvature at the origin of the 3-dimensional surface in R5 given by
yi=x\ + 2*1*2 + *2 + 5*1,
y2 = 3*1 + *1 + 2*2*3 + x\.
a. What is II (at the origin)?
b. What is the sectional curvature of the *i,*2-plane?
c. What is the sectional curvature of the plane *i + *2 = 0? of the plane *i + *2 + *3 = 0?
d. Give all the components of the Riemannian curvature tensor. Use them to recompute the answers to parts b and c.
e. Compute the Ricci and scalar curvatures.
5.2. Consider the vectorfield on R3: f = y2i + (* + z)j + *3k.
a. Compute its derivative at a general point in R3.
b. Compute its covariant derivative at (0,0,1) on the unit sphere.
5.3. Show that for the graph of a function /: R”-1 -* R,
„ Vf (l + |V/|2)A/-X/,/,/„
lvvTT[y7p (1 + |V/|2)3/2
where/, =dfldxi,fij = a2fldx,axl, V/s (/,,... div (p, q,. . .) =Pi + q2+ " and Af= div V/ = /u + /22 + • • • .
Since many analytic geometric quantities are intrinsic to a smooth m-dimensional surface S in R", the standard treatment avoids all reference to an ambient R". The surface S is defined as a topological manifold covered by compatible C°° coordinate charts, with a “Riemannian metric” g (any smooth positive definite matrix). This is not really a more general setting, since J. Nash [N] has proved that every such abstract Riemannian manifold can be isometrically embedded in some R". I suppose that it is a more natural setting, but the formulas get much more complicated.
So far we have seen one intrinsic quantity, the Gaussian curvature G of a 2-dimensional surface in R". We proved G intrinsic by deriving a formula for G in terms of the metric.
One may think of intrinsic Riemannian geometry as nothing but a huge collection of such formulas, thus proving intrinsic such quantities as Riemannian curvature, sectional curvature, and covari-ant derivatives. The standard approach uses these formulas as definitions. We have the advantage of having the simpler extrinsic definitions behind us. Formulas get much more complicated in intrinsic local coordinates.
In particular, complications arise because the local coordinates fail to be orthogonal, as in Figure 6.1. The «i-axis is not perpendicular to the level set {«i = 0}; or infinitesimally, the unit vector ex = d/dUi is not perpendicular to the level set {dux — 0}. Hence the
Figure 6.1. For nonorthogonal coordinates, the ux-axis is not perpendicular to the u2-axis (the level set {ui = 0}). Infinitesimally, the unit vector e1 = d/dui is not perpendicular to the level set {dui = 0}.
components of a vectorfield
X = (.X1, X2,..., Xm) = 2 X^i = 2 X1 —
and the components of a differential
(p = (<pb (p2,. . . , <pm) = 2 (pi du‘
behave very differently, under changes in coordinates for example. To emphasize the distinction, superscripts are used on the components of vector-like, or contravariant, tensors, and subscripts are used on the components of differential-like, or covariant, tensors.
Thus a vectorfield X has components X1. Its со variant derivative has components X\j, distinguished by the semicolon from the partial derivatives X\}- = dXllduj. As our first exercise in intrinsic Rieman-nian geometry, we will prove that the components of the со variant derivative are given by the formula
х\1 = х‘, + !,г'кхк, (i)
where T)k are the Christoffel symbols
Гу/с 2 g (§//,к + %lk,j %jk,l)i 2 /
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