By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This booklet will carry the wonder and enjoyable of arithmetic to the school room. It deals severe arithmetic in a full of life, reader-friendly kind. incorporated are workouts and plenty of figures illustrating the most ideas.

The first bankruptcy provides the geometry and topology of surfaces. between different issues, the authors speak about the PoincarÃ©-Hopf theorem on severe issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses quite a few elements of the idea that of measurement, together with the Peano curve and the PoincarÃ© strategy. additionally addressed is the constitution of third-dimensional manifolds. specifically, it's proved that the 3-dimensional sphere is the union of 2 doughnuts.

This is the 1st of 3 volumes originating from a sequence of lectures given via the authors at Kyoto collage (Japan).

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**Extra info for A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19)**

**Sample text**

For simplicity, we will only prove this formula for the special case where the zero set of K is discrete. Bott and Duistermaat-Heckman Formulas 32 We start with the simplest case. 1 If K has n o zeros o n M , then f o r any w E R*(M) which as dK-closed, one has JM w = 0. Proof. We use a method due t o Bismut [Bi2]. Let 9 E R1(M) be the one form on M such that for any X E r ( T M ) , ix9 = (XIK ) . 4), one then sees that (d i K ) e is dK-closed. The following lemma is due to Bismut [Bia]. Since d ~ is0 dK-closed, one verifies directly that from which one verifies directly, as dKw = 0, that 1, wexp ( - T d ~ 0 )- = 0.

Bismut, The Atiyah-Singer theorems: a probabilistic approach. J. Funct. Anal. 57 (1984), 56-99. -M. Bismut, Localization formulas, superconnections, and the index theorem for families. Commun. Math. Phys. 103 (1986)’ 127-166. [Bo] R. Bott, Vector fields and characteristic numbers. Michigan Math. J. 14 (1967)’ 231-244. [DH] J . J . Duistermaat and G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 (1982), 259-268. Addendum, 72 (1983), 153-158.

31) that when . . 29) holds. 28). 12 is thus completed. 2 Adiabatic Limit and the Bott Connection One may argue that from the geometric point of view, the connection VF' is also a natural connection on F'. In fact, by passing g T M t o its adiabatic limit, one sees that the underlying limit of VF' and the Bott connection are ultimately related. To be more precise, for any E > 0, let g T M be the metric on T M defined by oF' Let V T M >be€ the Levi-Civita connection of gTM+. Let VF+(resp. VFL>€) be the restriction of V T M +to F (resp.