Ivancevic_Applied-Diff-Geom

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Geometrical Path Integrals and Their Applications 1153Note that the indices of p l and α l n were lowered by the metric g ij not G ij .Recall the definition of the propagator〈X i (z, ¯z)X j (z ′ , ¯z ′ )〉 ≡ R(X i (z, ¯z)X j (z ′ , ¯z ′ )) − N(X i (z, ¯z)X j (z ′ , ¯z ′ )),(6.249)where R and N stand for the radial ordering and the normal orderingrespectively. We take a prescription for the normal ordering which pushesp i to the right and ˜x j to the left with respect to the zero–modes p i and ˜x j .It corresponds to considering the vacuum satisfyingp j |0〉 = α n,j |0〉 = 0 (n > 0), 〈0|α n,j = 0 (n < 0), (6.250)which is the standard prescription for calculating the propagator of themassless scalar field in 2D conformal field theory from the operator formalism.Making use of (6.249), (6.250) and techniques of the contour integration,it is easy to get the commutators[α n,i , α m,j ] = nδ n+m,0 G ij , [˜x i , p j ] = iδ i j,where the first equation holds for all integers with α 0,i ≡ √ 2α ′ p i . Theconstant D ij is written as α ′ D ij = −〈0|˜x i˜x j |0〉. Let us fix D ij as α ′ D ij =− i 2 θij , which is the convention taken in [Seiberg and Witten (1999)]. Thenthe coordinates ˜x i become noncommutative:[˜x i , ˜x j ] = iθ ij ,but the center of mass coordinates x i ≡ ˜x i + 1 2 θij p j can be seen to commuteeach other.Now we have the mode–expanded form of the string coordinates andthe commutation relations between the modes, which areX j (τ, σ) = x j + 2α ′ (G jk τ + 12πα ′ θjk (σ − π 2 ) )p k+ i √ 2α ∑ []′ 1n e−inτ G jk 1cos(nσ) − i2πα ′ θjk sin(nσ) α n,k ,n≠0[α n,i , α m,j ] = nδ n+m,0 G ij , [x i , p j ] = iδ i j,with all the other commutators vanishing.Also, due to the formula∞∑n=12n sin(n(σ + σ′ )) ={ π − σ − σ ′ , (σ + σ ′ ≠ 0, 2π)0, (σ + σ ′ = 0, 2π),
1154 Applied Differential Geometry: A Modern Introductionwe can see by a direct calculation that the end points of the string becomenoncommutative⎧⎨ iθ ij , (σ = σ ′ = 0)[X i (τ, σ), X j (τ, σ ′ )] = −iθ ij , (σ = σ ′ = π)⎩0, (otherwise).On the other hand, it is noted that the conjugate momenta have the modeexpansion identical with that in the Neumann case:P i (τ, σ) = 12πα ′ (g ij∂ τ − 2πα ′ B ij ∂ σ )X j (τ, σ)= 1 π p i + 1π √ ∑e −inτ cos(nσ)α n,i .2α ′Note that the relations (6.247) and (6.248) are in the same form as aT–duality transformation, although the correspondence is a formal sense,because we are not considering any compactification of space–time. Thegeneralized T–duality transformation, namely O(D, D)−transformation, isdefined byn≠0E ′ = (aE + b)(cE + d) −1 , (6.251)with a, b, c and d being D ×„D«real matrices. (D is the dimension ofa bspace–time.) The matrix h = is O(D, D) matrix, which satisfiesc dh T Jh = J, where J =( ) 0 1D.1 D 0The relations (6.247) and (6.248) correspond to the case of the inversiona = d = 0, b = c = 1 D .6.5.9.3 Construction of Overlap VerticesHere we construct Witten’s open string theory in the constant B−fieldbackground by obtaining the explicit formulas of the overlap vertices. As isunderstood from the fact that the action of the ghosts (6.239) contains nobackground fields, the ghost sector is not affected by turning on the B−fieldbackground. Thus we may consider the coordinate sector only. First, let us
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APPLIEDDIFFERENTIALGEOMETRYA Modern
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APPLIEDDIFFERENTIALGEOMETRYA Modern
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Dedicated to:Nitya, Atma and Kali
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PrefaceApplied Differential Geometr
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ixAcknowledgmentsThe authors wish t
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Glossary of Frequently Used Symbols
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ContentsPrefaceGlossary of Frequent
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Contentsxix2.1.5.2 Forces Acting on
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Contentsxxi3.6.3.4 Stokes Theorem a
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Contentsxxiii3.10.3.1 Basis of Lagr
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Contentsxxv3.17 Applied Unorthodox
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Contentsxxvii4.9.8.2 Geometrodynami
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Contentsxxix4.14.7.5 Monopole Conde
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Contentsxxxi5.12.3 Hawking-Penrose
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Contentsxxxiii6.5.2.4 Dimensional R
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Chapter 1IntroductionIn this introd
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Introduction 3Fig. 1.1 The four cha
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Introduction 5• Riemannian manifo
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Introduction 7charts.The atlas cont
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Introduction 9every point has a nei
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Introduction 11of curves include ci
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Introduction 13the variables define
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Introduction 15U i denotes one of t
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Introduction 17which varies smoothl
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Introduction 19interactions.1.1.5.2
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Introduction 21real number. This co
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Introduction 23Such a force is inde
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Introduction 25tons. In this formul
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Introduction 27vector-field are sol
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Introduction 29or simply connected)
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Introduction 311.1.8 Application: A
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Introduction 33theory itself existe
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Introduction 35mediate the forces.
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Introduction 37theory. 50 Fig. 1.3
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Introduction 39to describe a univer
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Introduction 41which have two disti
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Introduction 43first theory can be
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Introduction 45the Casimir effect,
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Introduction 47• External coordin
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Introduction 49a connection-base de
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Chapter 2Technical Preliminaries: T
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Technical Preliminaries: Tensors, A
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Chapter 3Applied Manifold Geometry3
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Applied Manifold Geometry 139where
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Applied Manifold Geometry 141In loc
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Applied Manifold Geometry 143on the
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Applied Manifold Geometry 145exist
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Applied Manifold Geometry 147then E
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Applied Manifold Geometry 1490D man
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Applied Manifold Geometry 151Fig. 3
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Applied Manifold Geometry 153the fo
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Applied Manifold Geometry 155contin
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Applied Manifold Geometry 157In oth
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Applied Manifold Geometry 159scalar
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Applied Manifold Geometry 1633.6 Te
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Applied Manifold Geometry 165Let ϕ
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Applied Manifold Geometry 1673.6.1.
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Applied Manifold Geometry 169or, if
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Applied Manifold Geometry 171If X i
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Applied Manifold Geometry 173along
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Applied Manifold Geometry 175is dua
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Applied Manifold Geometry 179(1) α
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Applied Manifold Geometry 181diagra
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Applied Manifold Geometry 183Clearl
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Applied Manifold Geometry 187duces,
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Applied Manifold Geometry 189This i
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Applied Manifold Geometry 191Hodge
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Applied Manifold Geometry 193The in
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Applied Manifold Geometry 195since
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Applied Manifold Geometry 197If a l
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Applied Manifold Geometry 199asGive
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Applied Manifold Geometry 201The co
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Applied Manifold Geometry 203at the
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Applied Manifold Geometry 205For ex
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Applied Manifold Geometry 207axis,
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Applied Manifold Geometry 209For ex
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Applied Manifold Geometry 211The sp
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Applied Manifold Geometry 213corres
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Applied Manifold Geometry 215corres
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Applied Manifold Geometry 217Specia
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Applied Manifold Geometry 219the co
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Applied Manifold Geometry 221moment
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Applied Manifold Geometry 223produc
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Applied Manifold Geometry 225has a
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Applied Manifold Geometry 2273.8.5
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Applied Manifold Geometry 229in whi
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Applied Manifold Geometry 231with
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Applied Manifold Geometry 233itly:(
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Applied Manifold Geometry 235play t
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Applied Manifold Geometry 237Some r
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Applied Manifold Geometry 239Basic
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Applied Manifold Geometry 241orthog
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Applied Manifold Geometry 243is adm
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Applied Manifold Geometry 245Root S
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Applied Manifold Geometry 247groups
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Applied Manifold Geometry 251second
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Applied Manifold Geometry 255be a v
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Applied Manifold Geometry 257Now, a
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Applied Manifold Geometry 263other
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Applied Manifold Geometry 267result
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Applied Manifold Geometry 2713.10 R
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Applied Manifold Geometry 273More p
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Applied Manifold Geometry 275Y (t 0
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Applied Manifold Geometry 277which
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Applied Manifold Geometry 279for an
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Applied Manifold Geometry 283Thus w
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Applied Manifold Geometry 287evolut
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Applied Manifold Geometry 293‘ren
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Applied Manifold Geometry 295mute;
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Applied Manifold Geometry 297Rather
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Applied Manifold Geometry 299of the
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Applied Manifold Geometry 301clidea
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Applied Manifold Geometry 303Second
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Applied Manifold Geometry 305i.e.,
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Applied Manifold Geometry 307invari
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Applied Manifold Geometry 309The Ba
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Applied Manifold Geometry 311Let us
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Applied Manifold Geometry 313all cu
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Applied Manifold Geometry 315ders.
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Applied Manifold Geometry 317this c
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Applied Manifold Geometry 319In par
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Applied Manifold Geometry 321As I p
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Applied Manifold Geometry 323the fo
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Applied Manifold Geometry 325per (1
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Applied Manifold Geometry 327Genera
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Applied Manifold Geometry 329The ma
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Applied Manifold Geometry 331pretat
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Applied Manifold Geometry 333byMDL
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Applied Manifold Geometry 3353.12 S
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Applied Manifold Geometry 337H 2 (M
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Applied Manifold Geometry 339bracke
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Applied Manifold Geometry 341Any so
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Applied Manifold Geometry 343All on
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Applied Manifold Geometry 345The ac
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Applied Manifold Geometry 347linear
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Applied Manifold Geometry 349such t
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Applied Manifold Geometry 351Let γ
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Applied Manifold Geometry 355chart
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Applied Manifold Geometry 357phase-
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Applied Manifold Geometry 359( ) 0
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Applied Manifold Geometry 361hetero
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Applied Manifold Geometry 363thus c
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Applied Manifold Geometry 365Action
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Applied Manifold Geometry 367and we
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Applied Manifold Geometry 369action
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Applied Manifold Geometry 371and we
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Applied Manifold Geometry 373by (J(
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Applied Manifold Geometry 375fields
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Applied Manifold Geometry 377Let Tx
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Applied Manifold Geometry 381The ve
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Applied Manifold Geometry 383and gi
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Applied Manifold Geometry 385The co
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Applied Manifold Geometry 387tor Ca
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Applied Manifold Geometry 3912 −
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Applied Manifold Geometry 393comple
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Applied Manifold Geometry 395F[C]
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Applied Manifold Geometry 397become
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Applied Manifold Geometry 399of fuz
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Applied Manifold Geometry 401In our
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Applied Manifold Geometry 403Biodyn
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Applied Manifold Geometry 405for T
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Applied Manifold Geometry 407A vari
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Applied Manifold Geometry 409such t
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Applied Manifold Geometry 411of neg
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Applied Manifold Geometry 415Follow
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Applied Manifold Geometry 417and J
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Applied Manifold Geometry 419This t
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Applied Manifold Geometry 421above,
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Applied Manifold Geometry 423As a L
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Applied Manifold Geometry 425Theore
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Applied Manifold Geometry 427real v
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Applied Manifold Geometry 429indepe
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Applied Manifold Geometry 431In ter
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Applied Manifold Geometry 433M, we
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Applied Manifold Geometry 435The cu
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Applied Manifold Geometry 437(iv) (
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. ..Applied Manifold Geometry 441in
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Applied Manifold Geometry 443⌋ is
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Applied Manifold Geometry 445Now, a
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Applied Manifold Geometry 447symmet
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Applied Manifold Geometry 449study
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Applied Manifold Geometry 451togeth
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Applied Manifold Geometry 453result
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Applied Manifold Geometry 455will b
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Applied Manifold Geometry 457g 11 =
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Applied Manifold Geometry 459is def
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Applied Manifold Geometry 461ordere
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Applied Manifold Geometry 463Now, r
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Applied Manifold Geometry 465Simila
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Applied Manifold Geometry 467which
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Applied Manifold Geometry 469From t
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Applied Manifold Geometry 475On the
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Applied Manifold Geometry 477ing di
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Applied Manifold Geometry 479Note t
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Applied Manifold Geometry 481Then w
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Applied Manifold Geometry 483consis
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Chapter 4Applied Bundle Geometry4.1
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Applied Bundle Geometry 487is calle
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Applied Bundle Geometry 489V and an
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Applied Bundle Geometry 4914.3 Vect
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Applied Bundle Geometry 493dimensio
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Applied Bundle Geometry 495orthogon
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Applied Bundle Geometry 497t, is a
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Applied Bundle Geometry 499Let T Y
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Applied Bundle Geometry 501In other
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Applied Bundle Geometry 503we shall
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Applied Bundle Geometry 505Pendulum
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Applied Bundle Geometry 507Using L
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Applied Bundle Geometry 509the isom
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Applied Bundle Geometry 511theory f
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Applied Bundle Geometry 513of the c
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Applied Bundle Geometry 515where Ω
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Applied Bundle Geometry 517nals, th
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Applied Bundle Geometry 519(1985)])
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Applied Bundle Geometry 521finite-d
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Applied Bundle Geometry 523and thes
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Applied Bundle Geometry 525Now, let
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Applied Bundle Geometry 527gravity
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Applied Bundle Geometry 529a Dp−b
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Applied Bundle Geometry 531A princi
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Applied Bundle Geometry 533is the r
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Applied Bundle Geometry 535Now, let
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Applied Bundle Geometry 537By cycli
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Applied Bundle Geometry 5394.9.2 Fe
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Applied Bundle Geometry 541This lin
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Applied Bundle Geometry 543(1) L g
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Applied Bundle Geometry 545and ω 0
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Applied Bundle Geometry 547isfiesω
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Applied Bundle Geometry 549of the c
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Applied Bundle Geometry 551Controll
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Applied Bundle Geometry 553Foliatio
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Applied Bundle Geometry 555ψ : (x,
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Applied Bundle Geometry 557from res
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Applied Bundle Geometry 559Motion a
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Applied Bundle Geometry 561where Z
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Applied Bundle Geometry 563Systems
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Applied Bundle Geometry 5653. Recal
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Applied Bundle Geometry 567Therefor
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Applied Bundle Geometry 569smooth m
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Applied Bundle Geometry 571state va
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Applied Bundle Geometry 573The geom
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Applied Bundle Geometry 575Y ∋ y
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Applied Bundle Geometry 577autonomo
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Applied Bundle Geometry 579(see 3.1
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Applied Bundle Geometry 581and init
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Applied Bundle Geometry 583potentia
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Applied Bundle Geometry 585effector
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Applied Bundle Geometry 587Fig. 4.9
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Applied Bundle Geometry 589physical
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Applied Bundle Geometry 591open Lio
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Applied Bundle Geometry 593et al. (
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Applied Bundle Geometry 595(2) Acti
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Applied Bundle Geometry 597˙Φ ≤
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Applied Bundle Geometry 599number o
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Applied Bundle Geometry 601where Ψ
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Applied Bundle Geometry 603ing to c
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Applied Bundle Geometry 605where [
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Applied Bundle Geometry 607interval
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Applied Bundle Geometry 609The foll
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Applied Bundle Geometry 611In parti
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Applied Bundle Geometry 613The pull
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Applied Bundle Geometry 615(2) δ
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Applied Bundle Geometry 617of unita
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Applied Bundle Geometry 619system o
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Applied Bundle Geometry 621of H pre
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Applied Bundle Geometry 623It follo
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Applied Bundle Geometry 625a produc
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Applied Bundle Geometry 627B Ham(M)
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Applied Bundle Geometry 629b.Let us
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Applied Bundle Geometry 631If P is
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Applied Bundle Geometry 633question
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Applied Bundle Geometry 635spheres
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Applied Bundle Geometry 637be thoug
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Applied Bundle Geometry 639of the b
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Applied Bundle Geometry 641in C 1 (
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Applied Bundle Geometry 643from a t
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Applied Bundle Geometry 645Now let
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Applied Bundle Geometry 647for t <
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Applied Bundle Geometry 649is a Ham
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Applied Bundle Geometry 651The foll
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Applied Bundle Geometry 653is the u
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Applied Bundle Geometry 655every no
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Applied Bundle Geometry 657P × P w
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Applied Bundle Geometry 659that the
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Applied Bundle Geometry 661extend t
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Applied Bundle Geometry 663identity
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Applied Bundle Geometry 665The mapw
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Applied Bundle Geometry 667It is a
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Applied Bundle Geometry 669I Q is i
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Applied Bundle Geometry 671whereC i
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Applied Bundle Geometry 673form Q.
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Applied Bundle Geometry 675τ( exp(
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Applied Bundle Geometry 6774.13.2.2
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Applied Bundle Geometry 679Therefor
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Applied Bundle Geometry 683⎛⎞co
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Applied Bundle Geometry 685Usually,
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Applied Bundle Geometry 687on the s
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Applied Bundle Geometry 6914.13.3 P
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Applied Bundle Geometry 693Since T
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Applied Bundle Geometry 695On the o
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Applied Bundle Geometry 697may writ
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Applied Bundle Geometry 699determin
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Applied Bundle Geometry 701section,
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Applied Bundle Geometry 703for deri
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Applied Bundle Geometry 705dictions
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Applied Bundle Geometry 707too sing
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Applied Bundle Geometry 709In (4.18
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Applied Bundle Geometry 713sis repr
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Applied Bundle Geometry 717a (linea
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Applied Bundle Geometry 719past few
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Applied Bundle Geometry 721global q
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Applied Bundle Geometry 7239. It ha
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Applied Bundle Geometry 725needs to
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Applied Bundle Geometry 727pear at
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Applied Bundle Geometry 729X, F A i
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Applied Bundle Geometry 731ifolds,
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Applied Bundle Geometry 733and not
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Applied Bundle Geometry 735under U(
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Applied Bundle Geometry 737N = 2 br
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Applied Bundle Geometry 739thusa
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Applied Bundle Geometry 741antisymm
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Applied Bundle Geometry 743see this
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Applied Bundle Geometry 745Coupling
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Applied Bundle Geometry 747acts as
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Applied Bundle Geometry 749good coo
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Applied Bundle Geometry 751Φ. Brea
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Applied Bundle Geometry 753equating
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Applied Bundle Geometry 755differen
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Applied Bundle Geometry 7574.14.10
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Applied Bundle Geometry 759deformed
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Applied Bundle Geometry 761phenomen
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Applied Bundle Geometry 763appears
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Applied Bundle Geometry 7654.14.10.
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Applied Bundle Geometry 767For J >
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Applied Bundle Geometry 769K 1/2
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Applied Bundle Geometry 771(i) The
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Applied Bundle Geometry 773by α =
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Applied Bundle Geometry 775(4.279)
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Applied Bundle Geometry 777Λ ⊗
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Applied Bundle Geometry 779coupling
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Applied Bundle Geometry 781where we
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Applied Bundle Geometry 783The posi
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Applied Bundle Geometry 785Although
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Applied Bundle Geometry 787Let us a
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Applied Bundle Geometry 789of the t
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Applied Bundle Geometry 791The Riem
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Applied Bundle Geometry 793To compl
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Applied Bundle Geometry 795Ω 1 :Ω
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Chapter 5Applied Jet GeometryModern
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Applied Jet Geometry 799x, with coe
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Applied Jet Geometry 801s i (x), as
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Applied Jet Geometry 803It is conve
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Applied Jet Geometry 805has the 1
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Applied Jet Geometry 807which split
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Applied Jet Geometry 809gives the h
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Applied Jet Geometry 811Furthermore
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Applied Jet Geometry 813This is a c
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Applied Jet Geometry 815Due to this
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Applied Jet Geometry 817The affine
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Applied Jet Geometry 819In particul
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Applied Jet Geometry 821Building on
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Applied Jet Geometry 823on J ∞ (X
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Applied Jet Geometry 825its tangent
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Applied Jet Geometry 827Fig. 5.3 Hi
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Applied Jet Geometry 829It follows
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Applied Jet Geometry 831affine, whi
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Applied Jet Geometry 833where Γ i
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Applied Jet Geometry 8355.6.5 Jacob
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Applied Jet Geometry 837is also the
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Applied Jet Geometry 839A generic m
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Applied Jet Geometry 841where ϑ eH
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Applied Jet Geometry 843Since E L|
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Applied Jet Geometry 845Thus, every
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Applied Jet Geometry 847along u. Ev
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Applied Jet Geometry 849equations c
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Applied Jet Geometry 851A Hamiltoni
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Applied Jet Geometry 853Connections
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Applied Jet Geometry 855Lagrangian
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Applied Jet Geometry 857on V ∗ M.
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Applied Jet Geometry 859contain no
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Applied Jet Geometry 861(ii) For an
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Applied Jet Geometry 8635.6.16 Lyap
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Applied Jet Geometry 865for any t >
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Applied Jet Geometry 867vertical co
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Applied Jet Geometry 869difficulty,
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Applied Jet Geometry 871the energy
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Applied Jet Geometry 873by the tran
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Applied Jet Geometry 875The conditi
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Applied Jet Geometry 877Let Γ = (M
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Applied Jet Geometry 879geometrical
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Applied Jet Geometry 881Let (M, I)
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Applied Jet Geometry 883so referrin
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Applied Jet Geometry 885The Poincar
Page 922 and 923:
Applied Jet Geometry 887where N n
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Applied Jet Geometry 889be a projec
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Applied Jet Geometry 891to differen
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Applied Jet Geometry 893for every c
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Applied Jet Geometry 895vector-fiel
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Applied Jet Geometry 897This is phr
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Applied Jet Geometry 899where a, b
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Applied Jet Geometry 901represent t
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Applied Jet Geometry 903̂L : J 1 (
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Applied Jet Geometry 905space J 1 (
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Applied Jet Geometry 907First of al
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Applied Jet Geometry 909Therefore,
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Applied Jet Geometry 911Recall that
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Applied Jet Geometry 9135.11 Applic
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Applied Jet Geometry 915the vector
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Applied Jet Geometry 9175.11.4 Gaug
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Applied Jet Geometry 9195.11.5 Lagr
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Applied Jet Geometry 921are represe
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Applied Jet Geometry 923which are t
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Applied Jet Geometry 925of this fun
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Applied Jet Geometry 927a principal
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Applied Jet Geometry 929Recall that
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Applied Jet Geometry 931Moreover, t
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Applied Jet Geometry 933other only
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Applied Jet Geometry 935In particul
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Applied Jet Geometry 937Note that,
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Applied Jet Geometry 939This equali
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Applied Jet Geometry 941Then, the s
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Applied Jet Geometry 943In the Lagr
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Applied Jet Geometry 945Let τ be a
Page 982 and 983:
Applied Jet Geometry 947Using (5.42
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Applied Jet Geometry 949associated
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Applied Jet Geometry 951parts,F α
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Applied Jet Geometry 953of the Hilb
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Applied Jet Geometry 955For example
Page 992 and 993:
Applied Jet Geometry 957the convent
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Applied Jet Geometry 959left ideal
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Applied Jet Geometry 961LX → Σ i
Page 998 and 999:
Applied Jet Geometry 963Levi-Civita
Page 1000 and 1001:
Applied Jet Geometry 965see this fr
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Applied Jet Geometry 967curves. And
Page 1004 and 1005:
Applied Jet Geometry 969One sees th
Page 1006 and 1007:
Applied Jet Geometry 971sums over a
Page 1008 and 1009:
Applied Jet Geometry 973all this to
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Applied Jet Geometry 975universe ev
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Applied Jet Geometry 977on Σ. If M
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Applied Jet Geometry 979(1) Lorentz
Page 1016 and 1017:
Applied Jet Geometry 981terpret thi
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Chapter 6Geometrical Path Integrals
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Geometrical Path Integrals and Thei
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Page 1138 and 1139: Geometrical Path Integrals and Thei
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BibliographyAblowitz, M.J., Clarkso
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Bibliography 1255equations. Springe
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Bibliography 1257connections, Phys.
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Bibliography 1259JHEP 0003:007.Bran
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Bibliography 1261Clementi, C. Petti
Page 1298 and 1299:
Bibliography 1263Domany, E., Van He
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Bibliography 1265geometrical signat
Page 1302 and 1303:
Bibliography 1267Bergmann theory of
Page 1304 and 1305:
Bibliography 1269to Complex Systems
Page 1306 and 1307:
Bibliography 1271Action for String
Page 1308 and 1309:
Bibliography 1273Ivancevic, V., Bea
Page 1310 and 1311:
Bibliography 1275Krener, A. (1984).
Page 1312 and 1313:
Bibliography 1277Li, M., Vitanyi, P
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Bibliography 1279Nature, 261(5560),
Page 1316 and 1317:
Bibliography 1281Lambert and R. Gol
Page 1318 and 1319:
Bibliography 1283dynamics of human
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Bibliography 1285Reshetikhin, N.Yu,
Page 1322 and 1323:
Bibliography 1287Rev. E, 58, 6333-6
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Bibliography 1289Stasheff, J.D. (19
Page 1326 and 1327:
Bibliography 1291matical Physics. S
Page 1328 and 1329:
Bibliography 1293Witten, E. (1991).
Page 1330 and 1331:
Index1−jet bundle, 8271−jet lif
Page 1332 and 1333:
Index 1297Chan-Paton factor, 1238Ch
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Index 1299Dirac-Born-Infeld theory,
Page 1336 and 1337:
Index 1301Gauss, 5Gauss map, 284Gau
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Index 1303isometric, 17isomorphic,
Page 1340 and 1341:
Index 1305Maxwell Lagrangian, 34Max
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Index 1307probability manifold, 333
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Index 1309super-space, 1215supercov
Page 1346:
Index 1311Yang-Mills theory, 39, 29
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