Classical topology and quantum states by Aiyalam P Balachandran, Giuseppe Marmo, Bo-Sture Skagerstam
By Aiyalam P Balachandran, Giuseppe Marmo, Bo-Sture Skagerstam
Balachandran, Marmo, Skagerstam, Stern. Classical topology and quantum states (WS, 1991)(ISBN 9810203292)(KA)(T)(375s)_PQgf_
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In particular by [BG01, eq. 16), Th,g T H M, g T X , ∇F ∗ ⊗o(T X) , gF ∗ ⊗o(T X) n+1 = (−1) Th,g T H M, g T X , ∇F , g F . 1) Of course, a similar identity holds for the Chern analytic torsion forms. 11 THE SECONDARY CLASSES FOR TWO METRICS Let g0F , g1F be two smooth g-invariant Hermitian metrics on F . 10], a secondary class hg ∇F , g0F , g1F ∈ Ω· (Mg ) /dΩ· (Mg ) was deﬁned such that dhg ∇F , g0F , g1F = hg ∇F , g1F − hg ∇F , g0F . 1) Let ∈ [0, 1] → g be a smooth family of Hermitian metrics which interpolates between g0F and g1F .
Equivalently, we get a canonical 1-form θ on M. Set ω = dM θ. 1) Then ω is a 2-form on M, which restricts to the canonical symplectic form on the ﬁbers of T ∗ X. 5], a horizontal vector bundle T H M is deﬁned, which is just the orthogonal vector bundle to T T ∗X in T M with respect to ω. 2) where ω , ω are the restrictions of ω to T T X, T M. 6], if T H is the restriction of T to T H M × T H M , V H ω H = p, T H . 3) Let T H be the ﬁberwise Hamiltonian vector ﬁeld whose associated Hamiltonian is just ω H .
Note here there should be no confusion between the ﬂat bundle F and the morphism F . Also f is a symmetric matrix, and, moreover, f = f F. 2), we observe that f deﬁnes a metric ∗ gT T X on T T ∗X given by gT T ∗ X = gT X 1|T X 1|T ∗ X ∗ 2g T X . 8) ∗ Then the volume form on T ∗ X which is attached to gT T X is just dvT ∗ X . 2) of T T ∗X. Then if U ∈ T T ∗X, U, U gT T ∗ X = π∗ U, π∗ U gT X + 2 π∗ U, pU + 2 pU, pU gT ∗ X . 9) 27 THE HYPOELLIPTIC LAPLACIAN ON THE COTANGENT BUNDLE Similarly, we will identify F to the gT T 1|T X 0 F = ∗ isometric involution of T T ∗X, X −1 2 gT X −1|T ∗ X .