By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This ebook will convey the sweetness and enjoyable of arithmetic to the study room. It bargains severe arithmetic in a full of life, reader-friendly type. incorporated are routines and plenty of figures illustrating the most suggestions.

The first bankruptcy provides the geometry and topology of surfaces. between different subject matters, the authors talk 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 a number of elements of the concept that of measurement, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of three-d manifolds. specifically, it's proved that the third-dimensional sphere is the union of 2 doughnuts.

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

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

**Sample text**

However, to get some understanding via analysis of vector bundles, it is necessary to introduce a generalized notion of function (reﬂecting the geometry of the vector bundle) to which we can apply the tools of analysis. , s maps a point in the base space into the ﬁbre over that point. S(X, E) will denote the S-sections of E over X. , S(U, E) = S(U, E|U ) [we shall also occasionally use the common notation (X, E) for sections, provided that there is no confusion as to which category we are dealing with].

We note that Ur,n (R) → Gr,n (R) is a real-analytic (and hence also differentiable) R-vector bundle and that Ur,n (C) → Gr,n (C) is a holomorphic vector bundle. The reason for the name “universal bundle” will be made more apparent later in this section. 7: Let π: E → X be an S-bundle and U an open subset of X. Then the restriction of E to U , denoted by E|U is the S-bundle π|π −1 (U ) : π −1 (U ) −→ U. , πE : E → X and πF : F → X. , f commutes with the projections and is a K-linear mapping when restricted to ﬁbres.

9: Let E −→X be an S-bundle. An S-submanifold F ⊂ E is said to be an S-subbundle of E if (a) F ∩ Ex is a vector subspace of Ex . , there exist local trivializations for E and F which are compatible as in the following diagram: E|U O i F |U ∼ / U × Kr O id × j ∼ / U × Ks, s ≤ r, where the map j is the natural inclusion mapping of K s as a subspace of K r and i is the inclusion of F in E. We shall frequently use the language of linear algebra in discussing homof morphisms of vector bundles. As an example, suppose that E −→F is a vector bundle homomorphism of K-vector bundles over a space X.