By Gerardo F. Torres del Castillo
This textbook explores the speculation in the back of differentiable manifolds and investigates quite a few physics purposes alongside the way in which. simple suggestions, corresponding to differentiable manifolds, differentiable mappings, tangent vectors, vector fields, and differential varieties, are in brief brought within the first 3 chapters. bankruptcy four supplies a concise creation to differential geometry wanted in next chapters. Chapters five and six offer fascinating functions to connections and Riemannian manifolds. Lie teams and Hamiltonian mechanics are heavily tested within the final chapters. integrated through the publication are a suite of workouts of various levels of trouble.
Differentiable Manifolds is meant for graduate scholars and researchers drawn to a theoretical physics method of the topic. necessities comprise multivariable calculus, linear algebra, differential equations, and a uncomplicated wisdom of analytical mechanics.
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Extra info for Differentiable Manifolds: A Theoretical Physics Approach
57) If t is a tensor field of type ( 0k ) on M and X1 , . . , Xk are k vector fields on M, owing to the linearity of tp in each of its arguments, for any function f : M → R, t (X1 , . . , f Xi , . . , Xk ) (p) = tp X1 (p), . . , (f Xi )(p), . . , Xk (p) = tp X1 (p), . . , f (p)Xi (p), . . , Xk (p) = f (p)tp X1 (p), . . , Xi (p), . . , Xk (p) = f (p) t (X1 , . . , Xi , . . , Xk ) (p), for p ∈ M, that is, t (X1 , . . , f Xi , . . , Xk ) = f t (X1 , . . , Xi , . . , Xk ), 1 ≤ i ≤ k.
53) a where the integral on the right-hand side is the Riemann integral of the real-valued function t → αC(t) (C (t)). As is well known, the value of C α depends on C only through its image and the direction in which these points are traversed. 9) we have b df = C b dfC(t) C (t) dt = a a Ct [f ] dt = b a d ∗ (C f ) dt dt = f C(b) − f C(a) . Hence, if C is a closed curve [that is, C(a) = C(b)], df = 0. 4 1-Forms and Tensor Fields 25 which is different from zero and, therefore, α is not the differential of some function defined on M [see also Guillemin and Pollack (1974), do Carmo (1994)].
Yk ), i=1 that is, k (£X t)(Y1 , . . , Yk ) = X t (Y1 , . . , Yk ) − t Y1 , . . , [X, Yi ], . . , Yk . 45). 30 Show that if X, Y ∈ X(M) and t ∈ Tk0 (M), then £X (£Y t) − £Y (£X t) = £[X,Y] t. 31 Show that if X ∈ X(M) and t ∈ Tk0 (M), then £X (X t) = X (£X t). 32 Let t be a differentiable tensor field of type ( kl ) on M. Assuming that the first k arguments of t are covectors and defining £X t by (£Xt)(α1 , . . , αk , Y1 , . . , Yl ) ≡ X t (α1 , . . , αk , Y1 , . . , Yl ) k − t (α1 , .