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By L. C. G. Rogers, David Williams

The booklet is intended to aid the study scholar succeed in the degree the place she or he can start either to imagine up and take on new difficulties and to learn the up to date literature throughout a large spectrum; and to cajole her or him that it's definitely worth the attempt.

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Buy Do Carmo’s (1976, 1992) two books, and start reading the one on curves and surfaces. If you are as bad learner as I am, do what I did, that is, all the exercises. Once you read about two dimensional surfaces and understand that curvature is a geometric name for a second order Taylor formula, you will have enough intuition to digest the abstract Riemannian manifolds — which really copy the classical theory of surfaces in R3 . After reading Do Carmo’s books, I found Chavel (1996) and some parts of Spivak (1970) most valuable.

From its definition, we infer that expp (ΩA,p) coincide with the set ωA,p = { q ∈ ΛI(A) : there exists a unique minimizing geodesic through q which meets DA orthogonally at p }. Set ωA = ωA,p . p∈DA On ωA , we can define a projection πA onto DA as follows. Any point q in ωA can be written in a unique way as q = expp (u) for (p, u) ∈ DA × Np DA . We set πA (q) = p. In other words, q is on a unique geodesic starting from DA and orthogonal to DA ; the projection of q on DA is the starting point of this geodesic in DA .

There are by now a few books on large deviations. Dembo and Zeitouni (1993) and Dupuis and Ellis (1997) are good starting points to the huge literature. From a different perspective, and restricted essentially to the univariate case, Jensen (1995) may be closer to what we are looking for here. Introducing the set Λc suggests that the variations of − log f are important. It has to do with the following essential remark. The negative exponential function is the only one — up to an asymptotic equivalence — for which integrating on an interval of length of order 1 produces a relative variation of order 1 on the integral.

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