By Animov Y.

Differential geometry is an actively constructing quarter of contemporary arithmetic. This quantity offers a classical method of the overall subject matters of the geometry of curves, together with the speculation of curves in n-dimensional Euclidean area. the writer investigates difficulties for designated periods of curves and offers the operating technique used to procure the stipulations for closed polygonal curves. The evidence of the Bakel-Werner theorem in stipulations of boundedness for curves with periodic curvature and torsion is additionally awarded. This quantity additionally highlights the contributions made by means of nice geometers. previous and current, to differential geometry and the topology of curves.

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36 4 Vector Fields, Diﬀerential Forms, and Derivatives The tangent space is isomorphic with an n-dimensional real vector space ∂ i through the canonical application θ : Tx X → Rm , θx (ξ i (x) ∂x i ) = (ξ ). The collection of all tangent spaces corresponding to all points of X is called tangent bundle and it is denoted as T X = ∪x∈X Tx X. By the property of overlapping and diﬀerentiability of charts in the atlas, all the tangent spaces on a manifold can be smoothly connected. Deﬁnition 6. A diﬀerentiable function v : X → Tx X is called a vector ﬁeld on the smooth manifold X.

A local comR2 pact linear topological space has ﬁnite dimension. , but we do not need these concepts in our book. Basically, they occur whenever we relax one of the three properties deﬁning compactness [11–13] (see Fig. 4). 3 Weierstrass–Stone Theorem How is it possible for the Taylor series to exist? That is, how is it possible to know all the values of a continuous function from just knowing a countable sequence of number, the coeﬃcients of the Taylor series. The answer is related to the separation axioms and it is the Weierstrass–Stone theorem.

V n , } be a ﬁnite set of n vector ﬁelds deﬁned on a smooth manifold X. We call integral submanifold of S a submanifold Y ⊂ X whose tangent space Tp Y is spanned by the system S at every point p ∈ N . The system at every point S is integrable if through every point p ∈ X there passes an integral submanifold. Deﬁnition 15. A ﬁnite system of vector ﬁelds S = {v 1 , v 2 , . . , if ∀p(x) ∈ X, ∀i, j = 1, . . , n n ckij (x)v k , [v i , v j ] = k=1 where ckij (x) are diﬀerentiable real functions on X, and [, ] is the Lie bracket deﬁned by the action (see Deﬁnition 9) of two smooth vector ﬁelds on functions f : M → R [v, w]f = v(w(f )) − w(v(f )).