By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This publication will carry the wonder and enjoyable of arithmetic to the school room. It bargains critical arithmetic in a full of life, reader-friendly sort. incorporated are routines and lots of figures illustrating the most ideas.

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 features of the idea that of size, together with the Peano curve and the Poincaré procedure. additionally addressed is the constitution of three-d 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 chain of lectures given via the authors at Kyoto collage (Japan).

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**Additional resources for A mathematical gift, 1, interplay between topology, functions, geometry, and algebra**

**Example text**

2, when Σ lies in a regular surface Γ , with parameterization X : U → Γ , we can think of Σ as being obtained by deforming an underlying curve in U . 32), we get |α (t )|2 = α (t ) · α (t ) = (s1 (t )∂ s1 X + s2 (t )∂ s2 X) · (s1 (t )∂ s1 X + s2 (t )∂ s2 X) = (s1 )2 ∂ s1 X · ∂ s1 X + 2s1 s2 ∂ s1 X · ∂ s2 X + (s2 )2 ∂ s2 X · ∂ s2 X, 33 34 Chapter 3. The Fundamental Forms of Differential Geometry where we suppressed the t notation in the last line. , d α2 = α (t ) · α (t ) d t 2 = d s12 ∂ s1 X · ∂ s1 X + 2d s1 d s2 ∂ s1 X · ∂ s2 X + d s22 ∂ s2 X · ∂ s2 X.

3. parabolic if κG = 0 but κ = 0 (one principal curvature is zero). 4. planar if κG = 0 and κ = 0 (both principal curvatures are zero). We say Γ is elliptic if all points in Γ are elliptic. A similar convention holds for the other surface types. 54 Chapter 3. 6. 62). (a) Elliptic surface: ρ = 1 produces a “sphere” like surface patch with κG > 0. (b) Parabolic surface: ρ = 0 produces a “cylinder” like surface patch with κG = 0 (but κ = 0). (c) Hyperbolic surface: ρ = −1 produces a “saddle” like surface patch with κG < 0.

5). 39) which is a symmetric matrix because h12 = h21 . h is a tensor that encodes the “curvature” of the surface—in fact, h is the matrix realization of the tensor. Just as with g , if X : n → m , then h is an n × n matrix. However, unlike g , h may not be positive definite. For example, if Γ is a plane in 3 , then h is the zero matrix (the reader can check this). 36) we can write h as a matrix-matrix product h = − (∇s ν)T (∇s X). 40) Similar to the first fundamental form, the second fundamental form is a quadratic form that is defined on the tangent space.