By Victor Andreevich Toponogov
The learn of curves and surfaces types an enormous a part of classical differential geometry. Differential Geometry of Curves and Surfaces: A Concise Guide offers conventional fabric during this box in addition to very important principles of Riemannian geometry. The reader is brought to curves, then to surfaces, and eventually to extra advanced themes. ordinary theoretical fabric is mixed with more challenging theorems and complicated difficulties, whereas protecting a transparent contrast among the 2 levels.
Key subject matters and features:
* Covers central techniques together with curves, surfaces, geodesics, and intrinsic geometry
* great fabric at the Aleksandrov international attitude comparability theorem, which the writer generalized for Riemannian manifolds (a end result referred to now because the celebrated Toponogov comparability Theorem, one of many cornerstones of contemporary Riemannian geometry)
* comprises many nontrivial and unique difficulties, a few with tricks and solutions
This rigorous exposition, with well-motivated issues, is perfect for complex undergraduate and first-year graduate scholars looking to input the fascinating global of geometry.
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Extra resources for Differential Geometry of Curves and Surfaces: A Concise Guide
The reader should consider the case in which C + (γ ) ∩ γ consists of exactly two points. 15. If γ is a closed convex curve of class C 2 , then the set C0 = C − (γ ) ∩ γ is connected. Solution. C0 is a closed set. Consequently, the set γ \ C0 is the union of open arcs. If C0 is not connected, the number of these arcs γ1 , γ2 , . . , γk is not smaller than 2. Two cases are possible: (1) The integral curvature of one of these arcs is smaller than π . (2) C0 consists of two diametrically opposite points P1 and P2 of the circle C − (γ ).
Prove that the length of any convex curve γ on the sphere S R is not greater than 2π R, and that equality holds if and only if γ is a great circle, or a biangle (lune) formed by two shortest paths between some points P and P ∗ . Solution. Let D be a convex region bounded by the curve γ . Consider the case that on γ there exist two diametrically opposite points P, P ∗ ∈ γ . Draw one of the shortest paths (P P ∗ )1 that is located inside of D. The existence of the shortest path (P P ∗ )1 follows from the deﬁnition of a convex 52 1.
Solution. If not, let C¯ be the closed semicircle of the circle C + (γ ) for which C¯ ∩ γ = ∅ holds. Denote by O the center of C + (γ ), and by R + its radius. Draw ¯ Denote by A1 and A2 the intersection points the diameter a through the ends of C. of the straight line containing the diameter a with γ . The points A1 and A2 divide γ into two arcs γ1 and γ2 . Denote by γ1 one of these arcs satisfying the property γ1 ∩ C + (γ ) = ∅. Let d0 = minγ (s)∈γ1 (Oγ (s) − R + ) > 0. 20. 14. to the line a from the center O inside of the semicircle C¯ and then mark off the line segment O O1 = d0 /2 = δ on it.