By Luther Pfahler Eisenhart

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**Extra resources for An introduction to differential geometry, with use of the tensor calculus**

**Example text**

It is well-known that a projective space P" is a differentiable manifold. 60) of a differentiable manifold. 71) for v = 0 and u = i , where i = 1,. . ,n, in the form: dub=w{ABj, where 0; = wj - 6;~:. 55) only in notation. 71), we find that de; = ejk AS: + (6;~;+ 6jw,O)A w,". 60), we observe that they coincide if Wjk = 6;w; + 6'wO 1 k' The latter relations prove that if a first order frame is fixed, the second order frames of a projective space P" depend on n parameters while on a general differentiable manifold they depend on n3 parameters.

A curve on a two-dimensional surface V 2 of a Euclidean space E3 is called asymptotic if its osculating planes coincide with the tangent planes to the surface V 2 or are undetermined (see for example, [BI 211, p. 52 or [Bl 501, p. 65). This definition is projectively invariant and can be generalized to the case where we have a submanifold of any dimension m in a projective space P". Namely, a curve 1 on a submanifold Vm is said to be asymptotic if its two-dimensional osculating plane at any of its points t belongs to the tangent subspace Ti1)to the submanifold Vm at this point or is undetermined.

M are not independent-they satisfy the sequence of quadratic relations: . . p w l . t m- 1 L p i o j l . 101) (see, for example, [HP 471). 102) (and many other formulas of this book), the square brackets enclosing some (or all) upper (or lower) indices denote the alternation with respect to t,he enclosed indices while the parentheses in the indices denote the symmetrization. For example, &I 1(+j- t j , i ) 2 = 12(+j+ t j , i ) - t[ijk] = L ( t i j k +tiki +tkij t j i k - t k j i - tikkj), 3' t ( i j k ) = L ( t i j k + tjki +tkij +tjik +t k j i + t i k j ) .

### An introduction to differential geometry, with use of the tensor calculus by Luther Pfahler Eisenhart

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