By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This booklet will carry the wonder and enjoyable of arithmetic to the study room. It deals critical arithmetic in a full of life, reader-friendly kind. integrated are routines and lots of figures illustrating the most thoughts.

The first bankruptcy provides the geometry and topology of surfaces. between different subject matters, the authors speak about the Poincaré-Hopf theorem on serious 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 numerous features of the concept that of measurement, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of 3-dimensional manifolds. specifically, it truly is proved that the three-d sphere is the union of 2 doughnuts.

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

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**Extra resources for A Mathematical Gift III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 23)**

**Sample text**

For more up-to-date treatment of differential topology, see Milnor (1965) and Hirsch (1976). One can construct an ambitious shopping list of books with which to start differential, and, more particularly, Riemannian geometry. Many of the more introductory books start with surfaces in Euclidean space and then deal with the exclusively intrinsic Riemannian geometry of surfaces. 2. Here, in addition to the references cited in the introduction to this chapter, we just note the influential two-volume treatise of Kobayashi–Nomizu (1969) on the general foundations – from connections through Riemannian metrics through curvature through the first level of specializations of areas in differential geometry, for example, curvature and geodesics, homogeneous spaces, K¨ahler manifolds, etc.

Proof. The functions t(τ ), ξ (τ ) are defined by setting t(τ ) = |(exp |B( p; ))−1 σ (τ )|, ξ (τ ) = (t(τ ))−1 (exp |B( p; ))−1 σ (τ ). 7), introduce geodesic spherical coordinates about p by defining V : [0, ) × S p → M by V (t, ξ ) = exp tξ. P1: IWV 0521853680c01 CB980/Chavel January 2, 2006 10:29 Char Count= 611 Riemannian Manifolds 24 As usual, set ∂t V = V∗ (∂t ) and (∂ξ V )η = V∗ (η) for η ∈ (S p )ξ . Then, for (t, ξ ) ∈ [0, ) × S p , we have ∂ξ V (t, ξ ) = t(exp p )∗|tξ tξ , as we argued previously in the proof of Gauss’s lemma.

1. If δ < , then there is nothing to prove; so assume δ ≥ and fix δ0 ∈ (0, ). Then, there exists p0 ∈ S( p; δ0 ) such that d( p0 , q) = d(S( p; δ0 ), q), that is, p0 is the point on S( p; δ0 ) closest to q. Our candidate geodesic γξ , therefore, is given by ξ the unit vector in M p determined by p and p0 , that is, ξ = (1/δ0 )(exp |B( p; ))−1 ( p0 ). 1) d(γξ (t), q) = δ − t. This will certainly prove the theorem. 1) is true for t = δ0 . Indeed, δ = d( p, q) ≤ d( p, p0 ) + d( p0 , q) = δ0 + d( p0 , q).