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Such points can be counted on some curves (see for example [Nu˜ no Ballesteros and Romero Fuster (1992, 1993); Sedykh (1992)]). One question of interest is how to approximate a surface in R3 by a developable surface. In [Izumiya and Otani (2015)] is considered the approximation of a surface by the developable surface tangent to it along a curve. page 22 October 12, 2015 10:9 9108 - Differential Geometry from a Singularity Theory Viewpoint 9789814590440 Chapter 2 Submanifolds of the Euclidean space In this chapter, we consider some aspects of the extrinsic geometry of a submanifold M of dimension n of the Euclidean space Rn+r , with r ≥ 1.

They form an n × n-symmetric matrix (hij ) and the second fundamental form can be written in matrix form IIp (w) = wT (hij (u))w. 7. The functions hij , i, j = 1, . . , n are called the coefficients of the second fundamental form and the matrix (hij ) is called the matrix of the second fundamental form. 2 (The Weingarten formula). The matrix of the shape operator Wp with respect to the basis B(x) of Tp M at p = x(u) is given by (hji (u)) with (hji (u)) = (hik (u))(g kj (u)) for (g ij (u)) = (gij (u))−1 , where the matrices, (gij (u)) and (hij (u)) are, respectively, those of the first and second fundamental forms of M at p.

Observe that the striction curve need not be a regular curve. We can now re-parametrise M by taking the striction curve σ as the base curve. The new parametrisation of M is given by y(t, u) = σ(t) + uβ(t) page 16 October 12, 2015 10:9 9108 - Differential Geometry from a Singularity Theory Viewpoint The case for the singularity theory approach 9789814590440 17 with t ∈ I and u ∈ R. Then, yt (t, u) = σ (t) + uβ (t), yu (t, u) = β(t), and yt × yu (t, u) = σ (t) × β(t) + uβ (t) × β(t). Since σ (t), β (t) = 0 and β(t), β (t) = 0, it follows that σ (t) × β(t) = λ(t)β (t) where the function λ : I → R is given by λ(t) = σ (t) × β(t), β (t) .