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Index

admissible lifting, 156“161, 165 ¬‚at model, 79“85, 91, 96“100,
a¬ne 102, 115“118
frame bundle, 58“65 conformal energy, 127
connection, 42, 52
fundamental forms, 59“65, 73
Cartan, 85, 91, 137
hypersurface, 37, 57“65, 73, 74
Levi-Civita, 22, 77, 87, 95, 141,
normal line, 60“63
142
transformation, x, 58“65
conservation law, ix, 14“21, 79
area functional, vii, 15, 24“35
“almost”, 34“35, 125
asymptotic curve, 184, 190
for cmc hypersurfaces, 32“34
B¨cklund transformation, xi, 23, 188,
a for harmonic maps, 166“168
for K = ’1 surfaces, 183“190
190“197
basic form, xii, 57 for minimal hypersurfaces, 27“
Betounes form, xi, 158“164 32
for wave equations, 118“127
Bianchi identity, 91“92
n+3
Bianchi, L., 193 for z = Cz n’1 , 123“127
n+2
Blaschke, W., 61 for ∆u = Cu n’2 , 110“118
“hidden”, 31
calibration, 145“149, 156 higher-order, 176“190
Cartan lemma, 24, 59, 60, 94, 108 proper, 15, 18“19
´
Cartan, E., ix, xi, 37 trivial, 15, 116, 188
Cauchy problem, 118“120 contact
Cauchy-Schwarz inequality, 129 form, 2, 7, 15, 16, 98, 102, 103,
characteristic cohomology, 4 124
characteristic system, 183“185 ideal, 3, 4, 10
Christodoulou, D., 118 line bundle, 4, 10, 16, 52, 123,
coframe bundle, 38 125
conformal manifold, ix, 1“8, 10, 39, 77,
bundle, 104 83, 97
curvature, 82 Cotten tensor, 92
frame bundle, 83, 85 covariant derivative, 90, 92, 94, 95,
group, 40, 79, 97, 101, 111, 139 109, 138“144, 165
inner-product, 80, 83, 136
Laplacian, 79, 80, 93“97, 102, Darboux theorem, 50
120, 138 decomposable form, 41, 53, 183
structure, x, 79“110, 133, 136“ density line bundle, 92“97, 120, 124,
140, 149 126, 138, 149

202
INDEX 203

dilation, 34, 84, 116, 121, 123 holomorphic curve, 31“32
divergence equivalence, 155, 156
ideal
algebraic, 2n, 154
energy, vii, 122“123, 127“129
di¬erential, 2, 9, 10, 158
of a map, 164, 165
independence condition, 2, 20
equivalence
integrable extension, xi, 185“188
of G-structures, 42
integral manifold, xii, 2, 4, 15
of conformal structures, 85“92
transverse, 12
of Lagrangians, 3
inverse problem, 1, 7, 10“14, 24, 39,
of Monge-Ampere systems, 39“
51
52
inversion, 84, 117, 121, 125
of Poincar´-Cartan forms, 65“
e
78, 102“110
Jacobi operator, ix, 145, 149“150
equivalence method, ix, 37“38, 41“
jets, viin, 2, 99“100, 103
53, 65“74, 79, 85“88, 102“
John, F., 129
110, 134
Euclidean K3 surface, 13
frame bundle, 21, 26, 28, 182 Killing vector ¬elds, 166
motion, 21“34, 78 Koszul complex, 144
space, vii, 21“35
Euler-Lagrange Lagrangian, vii“ix, 3“7, 155“164
equation, vii, viii, 7, 10, 72, 99, Lagrangian potential, 57“58, 73, 145
151, 156 Laplace equation, 139
system, ix, 1, 7“15, 40, 51, 133“ Legendre submanifold, 2“3, 7“11, 133“
149 145, 158
exterior di¬erential system, xii non-transverse, 23, 27, 196
transverse, 3, 9, 142
¬eld, 147“150 Legendre variation, 7“9, 134“149
¬rst variation, viii, 7, 133“134 Levine, H., 128n
Frobenius theorem, 55 Lorentz
functional, vii“ix, 3“7 conformal group, 119, 121, 126
frame, 81, 83, 98
Gauss curvature, 13, 41, 182“197 metric, 78, 119
Gauss map, 23 space, 80
generating function, 16“17 Lorentz boost, 121
Grassi, M., 155n, 163n
Grassmannian, 2, 13, 56, 142, 151 Maurer-Cartan
Green™s theorem, viii, 127“130 equation, 58, 82
G-structure, 37“38, 41“50, 85“88, 102 form, 80“82, 85, 100, 167
mean curvature, vii, 13, 24
H¨lder™s inequality, 130
o constant, 24, 27, 141
harmonic prescribed, x, xi, 38, 74, 77, 105n,
function, 29 133, 140“145
map, 164“168 minimal surface, ix, 15, 24“35, 78,
harmonic map, xi 145
Hodge-Lepage decomposition, 4 Minkowski space, 20, 118“127
204 INDEX

compacti¬ed, 119 Riemann curvature tensor, 22, 78,
89, 105, 137, 142
moment conditions, 31“34
Monge-Ampere system, 10“14, 99 Riemannian
hyperbolic, ix, 40“52, 182 frame bundle, 22, 77, 95, 141“
144, 164
moving frames, 59
Laplacian, 96, 144
multi-contact geometry, 151“158
manifold, 22, 74, 87
Noether™s theorem, ix, 1, 7, 10, 14“ metric, 77“78, 80, 95, 140
21, 28, 29, 79, 114, 115, rotation, 29, 34, 84, 116
163, 184 Rumin, M., 9n
non-degenerate
scalar curvature, 95, 106
1-form, 1
second fundamental form, 24, 26, 141,
a¬ne hypersurface, 59“60
183
functional, 9, 10, 15
of a map, 166
null vector, 80
second variation, ix, x, 78, 133“140
positive, 80, 97
for prescribed mean curvature,
Pfa¬ theorem, 2, 154 140“145
Pfa¬an system, xii semibasic form, xii, 38, 43, 57
Plateau problem, 31 simple foliation, 56, 78n, 97, 104
Pohoˇaev™s theorem, 117
z sine-Gordon equation, xi, 185“188,
Poincar´ lemma, 11, 50, 55
e 190“195
Poincar´-Cartan form, ix, 1, 6“11,
e space-like hypersurface, 78
133“147, 151“164 Spencer cohomology, 44, 87
almost-classical, 52“56, 65“67 stationary, vii“ix, 7“10, 134“149
conformally invariant, 97“118 stereographic projection, 79, 119
Euclidean-invariant, 25“26 Stokes™ theorem, 147“148
neo-classical, x, 37, 52“59, 65“ Strauss, W., 121n
67, 74“78, 98 stress-energy tensor, 166
de¬nite, 67“74, 102“110, 135 strong local minimum, 145“149
Poisson equation, x, 11“13, 79 summation convention, 68
conformally invariant, 110“118 symbol, 10, 162
non-linear, 96“97, 102“110, 118, symmetric form, 159, 165
188 symmetry, ix, 14“21, 78“85, 102, 111“
primitive, 4, 9n, 10“12, 39“40 113, 136
principal direction, 183 generalized, 188
prolongation hidden, 188
of a G-structure, x, 42, 85“88 symplectic
of a linear Lie algebra, 87 linear algebra, 4“6, 10, 11, 17“
of an EDS, 183 18, 50, 164

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