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| Berard P.H. — Spectral Geometry |
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| Предметный указатель |
Adjoint (formal) 119
Admissible functions 3—7 10 19
Asymptotic formula, Minakshisundaram and Pleijel 145
Asymptotic formula, Weyl 70 149
Boundary condition 4
Boundary condition, Dirichlet 3 13
Boundary condition, natural 3 13
Boundary condition, Neumann 3 13
Characterization see Eigenfunction Eigenvalue
Christoffel symbols 37
Closed (eigenvalue problem) see Eigenvalue
Co-area formula 80
Complete (Riemannian manifold) 25
Connection on tensor products 36
Connection, Levi Civita 35
Covariant derivative 35
Curvature, commuting derivatives and 42
Curvature, Gaussian 27
Curvature, Ricci 32 40
Curvature, scalar 40
Curvature, sectional 27 40
Curvature, space of constant 29
Curvature, tensor 39
Diameter 25
Direct (problem) 64 144
Dirichlet integral 14 55
Dirichlet, boundary condition see Boundary
Dirichlet, eigenvalue problem see Eigenvalue
Divergence of a vector-field 44
Divergence, Theorem see Theorem
Domination 165 168ff
Duhamel’s formula 99
Eigenfunction 13 52
Eigenfunction, characterization by  -density 62
Eigenvalue 13 52
Eigenvalue of Sturm — Liouville problem 9
Eigenvalue, asymptotic formula of Weyl 70 149
Eigenvalue, estimates of 73 74 112
Eigenvalue, multiplicity 9 53
Eigenvalue, problems (Closed, Dirichlet, Neumann) 52
Eigenvalue, variational characterization 60 61
Energy integral see Dirichlet integral
Exponential map 26
Faber — Krahn inequality 78
Geodesic 25 38
Gradient 42
Hausdorff distance 140
Heat, comparison theorem for the heat kernel 102 110
Heat, equation 98 145 149
Heat, isoperimetric inequalities and 101ff 108ff
Heat, kernel 99
Heat, kernel for a torus 100
Hessian (Riemannian) 41
Hodge (operator) 120
Hodge — de Rham operator 120
Hodge — de Rham theory 120
Hyperbolic space 21
Injectivity radius 71
Inverse problem, eigenvalue 64 144
Inverse problem, geometric 116
Isometry 22
Isoperimetric Cheeger’s constant 90
Isoperimetric estimator 84
Isoperimetric function 84 85
Isoperimetric inequality 82 84 85
Isoperimetric model space 86ff
Isospectral 146
Kato’s inequality 130 136 161 168 171 172ff
Laplace formulae for the Laplacian 45
Laplace — Beltrami operator (Laplacian) 15 45 120
Laplace, rough Laplacian 121 127 172
Laplacian see Laplace
Length (of curve) 24
| Multiplicity see Eigenvalue
Musical isomorphism 37
Neumann (eigenvalue problem) see Eigenvalue
Operator Laplace see Laplace
Operator positive 18 54
Operator positivity preserving 159 161
Operator symmetric 18 54
Paired, Pairing 164
Parallel, one-form 124
Parallel, translation 39
Parallel, vector-field 38
Partition function 100
Poisson summation formula 101 151
Projective space 23 29
Quotient see Rayleigh
Rayleigh — Ritz quotient 15 55
Riemannian canonical-measure 26
Riemannian distance 24
Riemannian fibre bundle 126
Riemannian invariant 23 47
Riemannian manifold 21 48
Riemannian metric 21
Riemannian product 22
Riemannian submersion 30
Scaling 24 26 34
Separation of variables 8 13
Sobolev inclusion theorem 51
Sobolev space 50
Solution of P.D.E., classical 17
Solution of P.D.E., weak 17
Space forms 29
Spectrum 146
Spectrum length 147
Sphere 22
Sturm — Liouville problem 9
Symmetrization 79 82 159ff 163ff 175ff
Theorem,  -bounds on eigenfunctions 114 139
Theorem, Beurling — Deny criterion 160 168 171
Theorem, Bishop comparison 34
Theorem, Bochner type 118 126 132 137 138
Theorem, Cheeger 92
Theorem, comparison 31 33
Theorem, comparison of heat kernels 102 110
Theorem, Daniel Meyer 131
Theorem, divergence 48
Theorem, Domination criterion 168 171
Theorem, E. Cartan 29
Theorem, estimates on eigenvalues 73 74 112
Theorem, estimates on isoperimetric constants 95
Theorem, existence of heat kernel 99
Theorem, Faber — Krahn 78 96
Theorem, fundamental 53
Theorem, Gauss — Bonnet 116
Theorem, Green 48
Theorem, Gromov compactness 140
Theorem, Gromov comparison 34
Theorem, Hopf — Rinow 25
Theorem, isoperimetric inequality 108
Theorem, Kato’s inequality 130 135 161
Theorem, lower-bounds on eigenvalues 112
Theorem, Minakshisundaram — Pleijel 145
Theorem, Myers 33
Theorem, on positivity preserving op. 161
Theorem, Toponogov 32
Theorem, vanishing 128
Theorem, Weyl 70 149
Torus 23 100
Unique continuation property 79
Wave equation 6 149
Weight of a Riemannian invariant 24 34
Weitzenboeck formula 121 127
Weyl asymptotic formula 70 149
Weyl conjecture 151
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