By Victor A. Toponogov (auth.), Vladimir Y. Rovenski (eds.)
The research of curves and surfaces kinds an enormous a part of classical differential geometry. Differential Geometry of Curves and Surfaces: A Concise Guide provides conventional fabric during this box besides very important rules of Riemannian geometry. The reader is brought to curves, then to surfaces, and at last to extra advanced issues. average theoretical fabric is mixed with tougher theorems and complicated difficulties, whereas retaining a transparent contrast among the 2 levels.
Key issues and features:
* Covers valuable strategies together with curves, surfaces, geodesics, and intrinsic geometry
* important 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 subject matters, is perfect for complex undergraduate and first-year graduate scholars looking to input the attention-grabbing global of geometry.
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Extra resources for Differential Geometry of Curves and Surfaces: A Concise Guide
Now let the points Q 1 and Q 2 have the properties given in the conditions of the problem. Then Q 1 Q 2 ≤ d = P1 P2 = b, but also Q 1 Q 2 ≥ b = d. From these inequalities it follows that Q 1 Q 2 = b, and hence Q 1 Q 2 is orthogonal to a1 and a2 . From the statements of the last problem it follows that at each point Q ∈ γ there is a unique point Q ∗ such that the tangent lines to γ at Q and Q ∗ are parallel. The points Q and Q ∗ are called diametrically opposite points on the curve γ . 27. If γ is a curve of class C 2 of the constant width a and the curvatures of γ at diametrically opposite points are equal, then γ is a circle of diameter a.
One should also ﬁnd the vector function r(s) using τ (s). 32) (τ 0 · ν 0 · β 0 ) = 1. 32) hold for any s. Introduce six new functions, ξ1 = τ (s), τ (s) , ξ2 = ν(s), ν(s) , ξ3 = β(s), β(s) , ξ4 = τ (s), ν(s) , ξ5 = τ (s), β(s) , ξ6 = ν(s), β(s) , and ﬁnd the ﬁrst derivatives of the functions ξi using the Frenet formulas: ⎧ ξ1 = τ , τ = 2 τ , τ = 2kξ4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ξ2 = 2 ν , ν = 2 −k τ + κ β, ν = −2kξ4 + 2κξ6 , ⎪ ⎪ ⎪ ⎪ ⎨ ξ = 2 β , β = 2 −κ ν, β = −2κξ6 , 3 ⎪ ξ4 = τ , ν + τ , ν = kξ2 − kξ1 + κξ5 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ξ5 = τ , β + τ , β = kξ6 − κξ4 , ⎪ ⎪ ⎪ ⎩ ξ6 = ν , β + ν, β = −κξ2 + κξ3 − kξ5 .
In this case, γ with the above-mentioned function of curvature would be a smooth regular curve having a ﬁnite number of arcs of class C 2 . To a point at which two arcs of class C 2 meet, correspond two values k− and k+ that are the left- and right-hand limits of the curvature function. We say that 24 1. Theory of Curves in Three-dimensional Euclidean Space and in the Plane the curvature of a curve γ at this point is not smaller than k0 (not greater than k0 ) if min(k− , k+ ) ≥ k0 (max(k− , k+ ) ≤ k0 ).
Differential Geometry of Curves and Surfaces: A Concise Guide by Victor A. Toponogov (auth.), Vladimir Y. Rovenski (eds.)