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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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[W] Steven Weinberg. Gravitation and Cosmology. New York: Wiley,
1972.
[Wu] G. Wulff. Zur Frage der Geschwindigneit des Wachsthums und der Auflösung der Krystullflachen. Zeitschrift für Krystallographi und Mineralogie 34 (1901), 449-530.
Solutions to Selected Exercises
3.1. a. 1/a, Ha, Ha2, Ha (or -1/a, -1/a, 1/a2, -2/a) b. 2a, 2b, Aab, 2(a + b)
H= 132 + 118 = 250,
G = 132 • 118 - 242 = 15,000,
d. Note that the -plane is not the tangent plane at 0, so use 3.5(3,4). H = 4V2, G = 6. Hence, k = 3V2, V2.
e. Switch variables and use 3.5(3,4). H = 0,
G =-1/(1+y2 sec2 zf = ~ 1/(1 + x2 + y2)2,
K — ± 1/(1 + y2 sec2 z) = ± 1/(1 + x2 + y2).
f. Use 3.5(3,4) and implicit differentiation.
5 • 34x2 + 10 • 2V + 13z2
(81*2 + 16y2 + z2)312
k =
H± V//2-4G 2
110 SOLUTIONS TO SELECTED EXERCISES
3.2. 2ir a sin c. (It is a circle of radius a sin c.)
3.3. x = (x,y,f(x,y)),n =
1)
H =
Vl +
C1 + fy)f*x - 2fxfyfx, + ( 1 + fl)fyy 1
(!+/?)(! + fi) - flfl Vl +fi+f:
Q _ fxxfyy fxy 1
1+fx+fU+fl+fy
3.4. xz = (/' cos Q,f sin Q, 1), xe= (—/sin 6,f cos d, 0), n = -(cos 6, sin 6, -/')(1 + /'2)-1/2 (inward), x*z = (/" cos 6,f" sin 6,0), xze = (-/' sin e,f cos 6,0), xee = (-/cos 0, -/sin 0,0),
rr -fr+m+r2) i , | i
/2(i+/'2) Vl +/'2 /Vl +/’2'
rfa
1 77 «*12
<P=0
.2 , n„2 (Lzjl _ , cn.,2l
3.5. A = J lira sin (pa d<p = 2m2(l — cos-^ = 27jt2 — -5 — r4 +
4.1. a. x = (*, y, x + 2y , 66x — 24xy + 59y ),
xx = (1,0,0,0), x2 = (0,1,0,0), xu = (0,0,2,132),
X12 = (0,0,0, -24), x22 = (0,0,4,118). Px«; = xiy.
By 4.2, H = (0,0,6, 250), G = 15,008.
Note that answers are sums of answers to Ex. 3.1(b, c).
b. x = (*, y, x2 - y2,2xy), Xi = (1,0,2x, 2y), x2 = (0,1, -2y, 2x), xu = (0,0,2,0), x12 = (0,0,0,2), x22 = (0,0, -2,0).
0
By 4.2, H = P--------------= 0. For G, we need Pxi;.
something
Since Xi, x2 are orthogonal,
\ Xu-Xi Xn-X2
F(Xll) = Xu--------2 Xt------T—x2
XI xi
{—Ax, Ay, 2,0)
1 + 4*2 + 4y2 ’
SOLUTIONS TO SELECTED EXERCISES 111
P(Xl2). LfcrfeM), P(X22).
1 + Ax + 4y 1 + 4x + 4y
G = -8(1 + 4x2 + 4y2)-3 = -8(1 + 4|z|2)-3.
4.2. x = (z,f(z)), with z = «1 + iu2.
*i = (1 >/'(*))» x2 = O', if'(z)), Xi • x2 = 0 (x2 = ixi), *ii = (0 x12 = (0, ifiz)), x22 = (0, -/"(z)).
_P(0)—
something
0. For G, we need Fxi;.
Since xl5 x2 span a complex subspace, F just projects onto the orthogonal complex subspace spanned by (-f(z), 1) G C2.
(~ff" f"\
Ptx ,-izMTÆl P(Xl2)- i + irp •
G=-2|r№ + l/12r3.
4.3. Calculating with formula 4.2(1) yields
h = Pv(i +1/,|2 +1/,!2)-1,
where
v = (1 + l/,|2)/,, - 2(/, ■/,)/,, + (1 + |/,|2)/,,
e R" 2 C R2 x R"
Clearly if v = 0, then H = 0 and the surface is minimal. On the other hand, if H = 0, v G ker F fl Rn~2 = {0}.
5.1. a.
"-[SI-
(2,6) (2,0) (0,0)
(2,0) (2,2) (0,2)
L(0,0) (0,2) (10,2) J
b. K(e, A e2) = (2,6) • (2,2) - (2,0) • (2,0) = 12.
c. One orthonormal basis is v = (1, —1,0)/V2, w = (0,0,1).
K = II(u, v) • II(w, w) - II(u, w) ■ II(u, w)
= (0,4) • (10,2) - (0, - V2) • (0, - V2) = 6.
One orthonormal basis for {xx + x2 + = 0} is u = ——h
V2
112 SOLUTIONS TO SELECTED EXERCISES
(1, 1, -2) vv = —7T-'
k = o.
d. Rl212 — 12, Rl213 — 12, Rl223 ~ 0, ^1313 — 32, /?1323 — 20,
R2323 - 20. Rest by symmetries.
Redoing b, c, and e, we have the following:
b. R1212 — 12.
c. i>x = -V2 = 1/V2, vv3 = 1, rest 0.
K = 2^1313 — 2Rl323 ~ 2.R2313 + 2.R2323 = 6.
Ui = — v2 = 1/V2, U3 = 0, vvx = w2 = 1/VS, vv3 = —2/V6.
= 12R1212 — — &R1312 + 3R1313 — 12R1221 +
12*M212 — 6*'•1213 — 6n1312 T 3*U313 — 12^1221 "r 6^1321 — 3Rl323 ~ T2R2112 + 6^2113 — 3R2313 + 12^2121 + 3R2323
= 1 - 2 - 2 + 32/3 + 1 - 2 - 20/3 + 1 - 2 - 20/3 + 1 + 20/3 = 0.
e. Ric =
44 20 O' 20 32 12 0 12 52
, R = 44 + 32 + 52 = 128.
5.2. Df= m°± a
Ldx dy dz_
" 0 2y 0"
= 1 0 1
_3x2 0 0_
= (0,2y, 0)1 + (1,0, l)j + (3x2,0,0)k.
Df( 0,0,1) =
0 0 i O' 1_0 ! i 0 0 0
= (0,0,0)1 + (1,0, l)j + (0,0,0)k.
Covariant derivative
ivative = U
Ll
5.3. Let g(*x, • • • >•*+) = xn ~ f(*l, • • • >xn-i). Then the graph of / is the level set {g = 0}. By 5.0(1),
Vg j. (~fu • • • , ~fn-1,1)
H = - div —-2- = - div / . ,
|Vg| Vi +/x + • • • +/„_x
- d- v/ - 9 1 d- v/
lvVi + |v/|2 ^„Vi + |v/|2 ,v Vi + |v/|2'
Symbol Index
X Euler characteristic, 67
Ck k times continuously differentiable, 11, 89
51 first variation, 13
ds2 metric, 18
Expp exponential map, 77
G Gauss curvature, 1, 13, 26, 71, 72
Gj Einstein tensor, 58
gij Riemannian metric, 18
T)k Christoffel symbols, 40
H mean curvature vector, 26, 27, 32
H scalar mean curvature, 1, 13, 19, 32
II second fundamental form, 12, 32, 34;
second fundamental tensor, 25, 31
K sectional curvature, 33, 42
k curvature vector, 5
k scalar curvature |k|, 1;
principal curvature, 13, 26, 34
Kg geodesic curvature vector, 47
n unit normal, 5, 11, 32;
Gauss map, 69, 71
d> norm, 89
R scalar curvature, 36, 42
Rjki Riemannian curvature tensor, 34, 41
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