Relating second order geometry of manifolds through projections and normal sections

We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\mathbb{R}^6$ (resp. $\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $\mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $\mathbb R^6$ and singular corank 1 surfaces in $\mathbb R^4$.


Introduction
The study of second order geometry of manifolds in Euclidean spaces dates as far back as Gauss.By second order geometry we refer to any geometrical aspects which can be captured by the second fundamental form, or, in modern terminology, by the 2-jet of a parametrisation of the manifold.Concepts such as elliptic/parabolic/hyperbolic points, normal curvature, asymptotic directions and some aspects of the contacts with hyperplanes and spheres are included in the study of second order geometry.
In his seminal paper [14], Little studied second order geometry of immersed manifolds in Euclidean spaces of dimensions greater than 3, in particular special attention was given to immersed surfaces in R 4 .He defined the second fundamental form and the curvature locus, which is an ellipse in this case.The curvature locus is the image in the normal space by the second fundamental form of the unitary tangent vectors.It can also be seen as the curvature vectors of normal hyperplane sections of the surface.The curvature locus is not an affine invariant but its topological type and its position with respect to the origin is an affine invariant.Besides, all the second order geometry is captured by this object.
The introduction of Singularity Theory techniques to study the differential geometry of manifolds in Euclidean spaces has given a great impulse to this subject in the last 20 years.There are many papers devoted to regular surfaces in R 4 such as [8,9,12,17,18,21,22,23], amongst others.For surfaces in R 5 [10,16,24] are good examples.In fact, there is a recent book which covers these topics ( [13]).The study of regular 3-manifolds in R 6 is also very recent.Here the curvature locus is a Veronese surface with many different topological types (see [6,7]).
Here the curvature locus is a parabola or a parabolic version of a Veronese surfaces.Curvature loci in general have been studied in [20], for example.
The aim of this paper is to relate the geometry of all these objects which have traditionally been studied separately.There is a natural relation between regular and singular objects.When projecting a regular n-manifold in R k along a tangent direction you obtain a singular n-manifold in R k−1 .On the other hand, taking normal hyperplane sections of the n-manifold gives a family of (n − 1)-manifolds in one dimension less.In Section 4 we establish a commutative diagram using projections and normal sections which induces a commutative diagram amongst the curvature loci with immersions and blow-ups.As a result of this we prove that the second order geometry of all these objects is related.This justifies known relations for projections when n = 2 and k = 4, for example, and motivates to look for further relations between the geometries of different manifolds, both regular and singular, in different Euclidean spaces.
Section 2 is devoted to preliminary results on the geometry of all the different objects appearing throughout the paper.In Section 3 we study normal sections of 3-manifolds both for the regular and singular cases and show that the curvature locus of a 3-manifold can be generated by the curvature loci of the surfaces obtained by normal sections.In particular we show how a Roman Steiner surface or a crosscap surface can be generated by ellipses.Using these sections we can recover some geometry of the 3-manifold by the topological types of the curvature loci of the sections.
In Section 5, inspired by the commutative diagram of Section 4, we define asymptotic directions for singular 3-manifolds in R 5 and relate them to asymptotic directions of regular 3-manifolds in R 6 and singular surfaces in R 4 .We prove that the direction of projection is asymptotic if and only if the singularity of the singular projection is not in the best A 2 -orbit.We then explain how this direction of projection becomes a null tangent direction in the singular projection and so justify the existence of infinite asymptotic directions in the singular case, which was not fully understood until now.
Aknowledgements: the authors would like to thank M. A. S. Ruas for useful conversations and constant encouragement.The second fundamental form of M 2 reg at a point p is defined by reg is the canonical projection on the normal space.We extend II p to the hole space in a unique way as a symmetric bilinear map.
Taking w = w 1 e 1 + w 2 e 2 ∈ T p M 2 reg , we can write the quadratic form where l i = f xx , e 2+i , m i = f xy , e 2+i and n i = f yy , e 2+i , for i = 1, . . ., k, are called the coefficients of the second fundamental form with respect to the frame above.The matrix of the second fundamental form with respect to the orthonormal frame above is given by Consider a point p ∈ M 2 reg and the unit circle S 1 in T p M 2 reg parametrised by θ ∈ [0, 2π].The curvature vectors η(θ) of the normal sections of M 2 reg by the hyperplane θ ⊕ N p M 2 reg form an ellipse in the normal space N p M 2 reg , called the curvature ellipse of M 2 reg at p, denoted by ∆ e , which is the same as the image of the map η : Moreover, if we write u = cos(θ)e 1 + sin(θ)e 2 ∈ S 1 , II p (u, u) = η(θ).

2.2.
Second order geometry of 3-manifolds in R n .Let M 3 reg be a 3-manifold in R 3+k , k ≥ 1, given locally as the image of the map f : , where f x = ∂f ∂x , etc.The orthonormal frame {e 1 , . . ., e k } is a frame of N p M 3 reg if the orientation of the frame {f x .fy , f z , e 1 , . . ., e k } coincides with the orientation of R 3+k .
The second fundamental form II p : T p M 3 reg × T p M 3 reg → N p M 3 reg is the bilinear map given by II p (v, w) = π 2 (d 2 f (v, w)), that projects the second derivative of f onto the normal space at p.The second fundamental form of M 3 reg at p along a normal vector field ν is the bilinear map II ν p : Given a point p ∈ M 3 reg , k ≥ 1, and a unit tangent vector v ∈ S 2 ⊂ T p M 3 reg , the normal (codimension 2) section of M 3 reg in the direction v is reg is an affine subspace of codimension 2 through p in R 3+k .We denote by η(v) the normal curvature vector of γ v in N p M 3 reg .When we vary v ∈ S 2 ⊂ T p M 3 reg , the vectors η(v) describe a surface in the normal space, called the curvature locus of M 3 reg at p and denoted by ∆ v The curvature locus can also be seen as the image of the unit tangent vectors at p via II(v, v).Lemma 2.1.[6] Taking spherical coordinates in S 2 ⊂ T p M 3 reg , we can parametrise the curvature locus of M 3 reg at p by η : . The affine hull of the curvature locus is denoted by Af f p and the linear subspace of N 1 p M 3 reg parallel to Af f p by E p .Also, a point p ∈ M 3 reg is said to be of type (M 3 reg ) i , i = 0, 1, . . ., 6 if dim N 1 p M 3 reg = i.The curvature locus of a regular 3-manifold in R 3+k can be seen as the image of the classical Veronese surface of order 2 via a convenient linear map.The expression of this surface is given by the image of the unit sphere in R 3 via the map ξ : ).For more details, see [6,7].
The next theorem describes the possible topological types of the curvature locus of a regular 3-manifold in R 6 .
Theorem 2.2.[6] The curvature locus at a point p of type (M 3 reg ) 3 in a 3-manifold M 3 reg ⊂ R 6 is isomorphic to one of the following: 1), a Cross-Cap surface (Figure 2), a Steiner surface of type 5 (Figure 3), a Cross-Cup surface (Figure 4), an ellipsoid or a (compact) cone.
∈ E p : An elliptic region, a triangle, a compact planar cone or a planar projection of type 1, 2 or 3 of the Veronese surface.Here, H(p) is the mean curvature vector.
The following refers to regular manifolds M n reg immersed in R 2n .For such objects, the definitions of tangent and normal spaces and second fundamental forms are analogues to the case of 3-manifolds in R 3+k (see [11] for details).The curvature Veronese submanifold of M n reg at p is the image of the unit tangent sphere S n−1 ⊂ T p M n reg via the second fundamental form.Let {e 1 , . . ., e n } and {v 1 , . . ., v n } be basis for T p M n reg and N p M n reg , respectively.For each u ∈ T p M n reg , A(u) denotes the n×n matrix such that A(u) ij = II v i (e j , u) = II(e j , u), v i .
Theorem 2.3.[11] Let f : M n reg → R 2n be an immersion and let u be a unit tangent vector at p ∈ M n reg .The following are equivalent: (i) The vector u satisfies det(A(u)) = 0; (ii) There exists a unit normal vector v such that II v (u, •) = 0; (iii) There exists a height function h v (x) = v, x such that h•f has a degenerate (non Morse) singularity with h, d 2 f u = 0; (iv) The vector II(u, u) is tangent to the curvature Veronese submanifold at II(u, u), or the curvature Veronese submanifold has a singularity at u.
sing be a corank 1 surface in R n , n = 3, 4, at p.The singular surface M 2 sing will be taken as the image of a smooth map g : M → R n , where M is a smooth regular surface and q ∈ M is a corank 1 point of g such that g(q) = p.Also, consider φ : U → R 2 a local coordinate system defined in an open neighbourhood U of q at M .Using this construction, we may consider a local parametrisation f = g • φ −1 of M at p (see the diagram below).
The tangent line of M 2 sing at p, T p M 2 sing , is given by Im dg q , where dg q : T q M → T p R 4 is the differential map of g at q. Thus, the normal space of M 2 sing at p, N p M 2 sing , is the subspace satisfying The second fundamental form of M 2 sing at p, II : and we extend it to the whole space in a unique way as a symmetric bilinear map.
Given a normal vector ν ∈ N p M 2 sing , we define the second fundamental form along ν, II ν : Given u = α∂ x + β∂ y ∈ T q M and fixing an orthonormal frame {ν 1 , . . ., Moreover, the second fundamental form is represented by the matrix of coefficients Let C q ⊂ T q M be the subset of unit tangent vectors and let η : C q → N p M be the map given by η(u) = II(u, u).The curvature parabola of M 2 sing at p, denoted by ∆ p , is the subset η(C q ).
The curvature parabola is a plane curve for both cases, n = 3 and n = 4, and it can degenerate into a half-line, a line or even a point.This special curve plays a similar role as the curvature ellipse does for regular surfaces.Therefore, it contains information about the second order geometry of the surface.
For the case n = 3, there is no doubt about which plane contains ∆ p , since the normal space is a plane.However, if n = 4, the normal space has dimension 3 and if ∆ p degenerates, we need to be careful.Let M 2 sing ⊂ R 4 be a corank 1 surface at p.The minimal affine space which contains the curvature parabola is denoted by Af f p .The plane denoted by E p is the vector space: parallel to Af f p when ∆ p is a non degenerate parabola, the plane through p that contains Af f p when ∆ p is a non radial half-line or a non radial line and any plane through p that contains Af f p when ∆ p is a radial half-line, a radial line or a point.
A non zero direction u ∈ T q M is called asymptotic if there is a non zero vector ν ∈ N p M 2 sing (for n = 3) or ν ∈ E p (for n = 4) such that II ν (u, v) = II(u, v), ν = 0 for all v ∈ T q M .Moreover, in such case, we say that ν is a binormal direction.
For the case n = 4, the normal directions ν ∈ N p M 2 sing which are not in the plane E p but also satisfy the condition II ν (u, v) = II(u, v), ν = 0, are called degenerate directions.The subset of degenerate directions in N p M 2 sing is a cone and the binormal direcitons are those in the intersection of this cone with E p .
2.4.Corank 1 3-manifolds in R 5 .In [5], the authors dedicate themselves to the study of singular corank 1 3-manifolds in R 5 , inspired by [4,15].In that paper, they define the fundamental forms, the curvature locus and also investigate some aspects of the second order geometry of those manifolds.Let M 3 sing ⊂ R 5 be a 3-manifold with a singularity of corank 1 at p ∈ M .The construction here is the same as for singular surfaces.We assume that M 3 sing is the image of a smooth map g : M → R 5 , where M is a smooth regular 3-manifold and q ∈ M is a singular corank 1 point of g such that g(q) = p.Taking φ : U → R 3 defined on some open neighbourhood U of q in M , we say that f = g • φ −1 is a local parametrisation of M 3 sing at p.The following definitions are analogous to the ones presented before: tangent space (T p M 3 sing ), normal space (N p M 3 sing ) and first fundamental form, I : T q M × T q M → R. Taking the frame {∂x, ∂y, ∂z} of T q M , the coefficients of the first fundamental form with respect to φ are: Notice that if u = a∂x + b∂y + c∂z then The second fundamental form of M 3 sing at p is the map I : given by where π 2 : T p R 4 → N p M 3 sing is the orthogonal projection and they are all evaluated in φ(q) and we extend II to T q M × T q M in a unique way as a symmetric bilinear map.
Given a normal vector ν ∈ N p M , the second fundamental form of M 3 sing at p along ν, II ν : The coefficients of II ν in terms of local coordinates (x, y, z) are: and the partial derivatives are all evaluated at φ(q).For a fixed orthonormal frame {ν 1 , ν 2 , ν 3 } of N p M 3 sing , the quadratic form associated to the second fundamental form is and the above coefficients calculated at q. Furthermore, in terms of the chosen frame, the second fundamental form can be represented by the following 3 × 6 matrix of coefficients: Let C q be the subset of unit vectors of T q M and let η : C q → N p M 3 sing be the map given by η(u) = II(u, u).We define the curvature locus of M 3 sing at p, which we shall denote by ∆ cv , as the subset η(C q ).Using suitable change of coordinates and rotations, we can write f (x, y, z) = (x, y, f 1 (x, y, z), f 2 (x, y, z), f 3 (x, y, z)), with (f i ) x = (f i ) y = (f i ) z = 0 at φ(q), for i = 1, 2, 3. Hence, the coefficients of the first fundamental form are E = G = 1 and F = H = I = J = 0. Furthermore, given a unit tangent vector u ∈ C q and writing u = x∂x + y∂y + z∂z, since x 2 E xx (q) + 2xyE xy (q) + y 2 E yy (q) + z 2 E zz (q) + 2xzE xz (q) + 2yzE yz (q) = 1 we have x 2 + y 2 = 1, that is, C q is a unit cylinder parallel to the z-axis.Fixing an orthonormal frame is a parametrisation for curvature locus ∆ cv , where x 2 + y 2 = 1.

Normal sections
Consider M 3 reg ⊂ R 3+k , k ≥ 1 a regular 3-manifold (resp.M 3 sing ⊂ R 5 a singular corank 1 3-manifold).Let u be a tangent direction in T p M 3 reg (resp.T p M 3 sing ) and {u = 0} the hyperplane in R 3+k (resp.R 5 ) orthogonal to u.The normal section of M 3 reg along u is a regular surface ).In view of this, one may ask whether there is a relation between the curvature locus of M 3 reg ⊂ R 3+k at p and the curvature ellipse of M 2 reg at p (resp. the curvature locus of M 3 sing ⊂ R 5 at p and the curvature parabola of M 2 sing at the same point).The answer to this questions is yes in both cases.Nevertheless, the cases will be treated separately, since the proof of the singular case is more delicate.Theorem 3.1.Let M 3 reg ⊂ R 3+k , k ≥ 1, a regular 3-manifold and p ∈ M 3 reg .The curvature locus of M 3 reg at p is generated by the union of the curvature ellipses at p of the regular surfaces in R 2+k given by the normal sections along the tangent directions of M 3 reg .
Proof.Assume, without loss of generality, that p is the origin.Take a parametrisation of M 3 reg in the Monge form f : (R 3 , 0) → (R 3+k , 0), with f (x, y, z) = (x, y, z, f 1 (x, y, z), . . ., f k (x, y, z)), reg be a non-zero vector and α 1 , α 2 , α 3 ∈ R not all zero such that u = α 1 X + α 2 Y + α 3 Z and α 2 1 + α 2 2 + α 2 3 = 1.Consider the normal section given by the (2 + k)-space generated by {α The tangent plane T p M 2 reg is such that its subset of unit vectors S 1 is also a subset of S 2 ⊂ T p M 3 reg , since the curvature locus of M 3 reg is the image, via second fundamental form, of S 2 and its restriction to T p M 2 reg is precisely the second fundamental form of M 2 reg at p. Therefore the curvature ellipse of M 2 reg at p is contained in the curvature locus of M 3 reg at p. Finally, varying u in S 2 ⊂ T p M 3 reg , we obtain all possible normal sections and the corresponding unit circles S 1 cover the sphere S 2 .Hence, the curvature locus of M 3 reg is given by the union of the curvature ellipses.
At the origin p, its curvature locus is a Roman Steiner surface.The normal sections given by {X = 0}, {Y = 0} and {Z = 0}, are regular surfaces whose curvature ellipses at p are, respectively: where θ ∈ [0, 2π].In all the cases, the curvature ellipse is a segment which corresponds to the double point curve of the Roman Steiner surface.The normal sections {X = Y }, {X = Z} and {Y = Z}, after changes of coordinates in the source and rotations in the tangent spaces of the surfaces in R 5 , provide us, respectively, the following curvature ellipses: 2 sin(θ) 2 ), where θ ∈ [0, 2π].This time, all curves are non degenerate ellipses.Figure 5 shows the curvature ellipses on the Roman Steiner surface.It seems Steiner himself already knew how to generate the Roman surface by ellipses (see [1]).However, all his ellipses pass through a "pole" whereas all of the ellipses obtained here pass through the triple point.(ii) Consider M 3 reg ⊂ R 5 given by f (x, y, z) = (x, y, z, x 2 + z 2 , xy).Taking coordinates (X, Y, Z, W, T ) in R 5 , its curvature locus at the origin p is an elliptic region contained in the normal plane {W, T }, with center at (1, 0) and radius 1. Table 1, shows some curvature ellipses of regular surfaces Table 1.Curvature ellipses on the elliptic region.

Normal section Parametrisation of the curvature ellipse
Type given by normal sections.Here, θ ∈ [0, 2π]. Figure 6 shows the curves in Table 1.
Although it was known that the Roman Steiner surface could be generated by ellipses, geometrically speaking this is not so obvious for the cross-cap, the Steiner Type 5 or the cross-cup surface.
sing ⊂ R 5 a singular corank 1 3-manifold.The curvature locus of M 3 sing at p is generated by the union of the curvature parabolas at p of the singular surfaces in R 4 given by the normal sections along the tangent directions of M 3 sing .Proof.Consider w ∈ T p M 3 sing a non zero vector.Here, (dg q ) −1 (w) ⊂ T q M is a plane which contains the subset ker(dg q ), where g is the corank 1 map at q used in the initial construction, where g(q) = p.Hence, the subset C q = (dg q ) −1 (w) ∩ C q is a pair of lines contained in the unit cylinder C q and such that η q (C q ) is the curvature parabola at p of the singular surface contained in the 4-space given by the normal section {w = 0}.Besides, the curvature parabola is a subset of the curvature locus of M 3 sing .The second fundamental form of M 3 sing restricted to (dg q ) −1 (w) ⊂ T q M is precisely the second fundamental form of the singular surface M 2 sing ⊂ R 4 .Figure 7 shows the previous construction.Varying w ∈ T p M 3 sing , we obtain the cylinder C q in T q M , therefore, the curvature locus of the 3-manifold: since each normal section induces two lines which cover the cylinder when varying the normal section, the curvature locus of M 3 sing at p is generated by the reunion of these curves.(i) Let M 3 sing ⊂ R 5 be the singular 3-manifold at the origin p locally given by f (x, y, z) = (x, y, x 2 − 2yz, y 2 − 2xz, z 2 − 2xy) whose curvature locus ∆ cv at p is sing : α 2 + β 2 = 1}, showed in Figure 8.The normal section given by {X = 0}, is the ) is such that its curvature parabola is also a non degenerated parabola, η(z) = (2, −4z, 2z 2 ).Taking the normal section {X +aY = 0}, where a = 0, after changes of coordinates in the source and isometries in the target, we obtain the singular surface given by 0, y, a 2 √ a 2 + 1y 2 − 2(a 2 + 1)yz (a 2 + 1) a non degenerate parabola for a ∈ R. Figure 9, shows some of the curvature parabolas in the curvature locus.(ii) Let M 3 sing ⊂ R 5 be locally parametrised by f (x, y, z) = (x, y, z 2 , xz, 0).The curvature locus ∆ cv at the origin p is the subset , a planar parabolic region, as in Figure 10.The normal section given by {X = 0} is the corank 1 surface parametrised by f (y, z) = (y, z 2 , 0, 0), whose curvature parabola is the half-line η(z) = (2z 2 , 0, 0).The remaining normal sections given by {Y + aX = 0}, where a ∈ R are corank 1 surfaces parametrised by fa (x, z) = (x, −ax, z 2 , xz, 0), that can be written (after  The curvature parabola's topological type of a corank 1 surface M 2 sing ⊂ R n , n = 3, 4 is a complete invariant for the A 2 -classification of 2-jets in Σ 1 J 2 (2, n), as shown in [4,15].The curvature locus of a corank 1 3-manifold in R 5 does not have the same property: the curvature locus of the 3-manifold given by g(x, y, z) = (x, y, xz, yz, z 2 ) at the origin p is the paraboloid ∆ p = {(0, 0, 2ac, 2bc, 2c 2 ) | a 2 + b 2 = 1}, but the 3-manifold given by f (x, y, z) = (x, y, xz + y 2 , yz, z 2 ), which satisfies j 2 f (0) ∼ A 2 (x, y, xz, yz, z 2 ) has the curvature locus at the origin ∆ cv = {(0, 0, 2b 2 + 2ac, 2bc, 2c 2 ) | a 2 + b 2 = 1}, which is not a paraboloid, as shown in [5].However, the topological type of the curvature parabolas of the normal sections gives necessary conditions for the A 2 -orbits of the 3-manifold's parametrisation.Theorem 3.5.Let M 3 sing ⊂ R 5 be a corank 1 3-manifold at p ∈ M 3 sing .We assume p the origin and denote by j 2 f (0) the 2-jet of a local parametrisation f : (R 3 , 0) → (R 5 , 0) of M 3 sing .The following holds: (i) j 2 f (0) ∼ A 2 (x, y, xz, yz, z 2 ) ⇒ ∆ cv is generated exclusively by non degenerate parabolas; (ii) j 2 f (0) ∼ A 2 (x, y, z 2 , xz, 0) ⇒ ∆ cv is generated by non degenerate parabolas and a half-line; (iii) j 2 f (0) ∼ A 2 (x, y, xz, yz, 0) ⇒ ∆ cv is generated exclusively by lines; (iv) j 2 f (0) ∼ A 2 (x, y, z 2 , 0, 0) ⇒ ∆ cv is generated exclusively by half-lines; (v) j 2 f (0) ∼ A 2 (x, y, xz, 0, 0) ⇒ ∆ cv is generated by lines and a point; (vi) j 2 f (0) ∼ A 2 (x, y, 0, 0, 0) ⇒ ∆ cv is generated exclusively by points.
Proof.Since the proofs of all cases are similar, we shall present only the first case.In [5], the authors proved that if where c 4 > 0 and b 6 = 0. Here, R 2 denotes the group of 2-jets of diffeomorphisms from (R 3 , 0) to (R 3 , 0) and O( 5) is the group of linear isometries of R 5 .
Consider the normal section given by {Y + αX = 0}, where α ∈ R − {0}, locally parametrised by By a rotation of angle θ = arctan(α) in the target and the change of coordinates in the source, (x, z) → ( The parametrisation of the normal section in the 4-space XZW T , is such that its 2jet is A 2 -equivalent to (x, xz, z 2 , 0), since the coefficient of z 2 is not zero.Hence, by Theorem 3.6 in [4], the curvature parabola of the normal section is a non degenerate parabola for all α = 0. Finally, the normal sections given by {X = 0} and {Y = 0} are singular surfaces parametrised, respectively, by (y, z) → (0, y, a 3 y 2 + a 6 yz, b 3 y 2 + b 6 yz, c 3 y 2 + c 4 z 2 + c 6 yz), and and the 2-jets of both of them are A 2 -equivalent to (x, xy, y 2 , 0).Once again, by Theorem 3.6 in [4], the curvature parabolas are non degenerate parabolas.Therefore, ∆ cv is obtained exclusively by non degenerate parabolas.
The converse of Theorem 3.5, nevertheless, is not true.The curvature locus of M 3 sing given by f (x, y, z) = (x, y, z 2 , xz, 0) at the origin p, as in Example 3.4, is a planar region that can be seen as the union of only non degenerate parabolas.and by E ij := f i , f j the coefficients of the first fundamental form.Consider a unit tangent vector u = n i=1 a i f i ∈ T p M n reg , then I(u, u) = n i=1 a 2 i E ii + 2 1≤i<j≤n a i a j E ij = 1.Consider the projection in the direction u, P u = f − f, u u.The coefficients of the first fundamental form for the singular projection are E P ii = P u i , P u i = f i − f i , u u, f i − f i , u u = E ii − ( f i , u ) 2 and similarly E P ij = E ij − f i , u f j , u .So the first fundamental form of the singular projection applied to u is then it is A 2 -equivalent to one of the following orbits: (x, y, xz, yz, z 2 ), (x, y, z 2 , xz, 0), (x, y, xz, yz, 0), (x, y, z 2 , 0, 0), (x, y, xz, 0, 0), (x, y, 0, 0, 0).Furthermore, they show that it is in the best For simplicity we take Monge forms and the direction of projection u ∈ T p M 3 reg to be (0, 0, 1).Notice that in this setting u is the null tangent direction in T q M .Proposition 5.8.The direction u = (0, 0, 1) ∈ T p M 3 reg is an asymptotic direction if and only if j 2 P u (0) is not in the orbit (x, y, xz, yz, z 2 ), where P u stands for the projection of M 3 reg in the direction u.

2 . 2 . 1 .
The geometry of regular and singular surfaces and 3-manifolds in Euclidean spaces Regular surfaces in Euclidean spaces.Given a smooth surface M 2 reg ⊂ R 2+k , k ≥ 2 and f : U → R 2+k a local parametrisation of S with U ⊂ R 2 an open subset, let {e 1 , . . ., e 2+k } be an orthonormal frame of R 2+k such that at any u ∈ U , {e 1 (u), e 2 (u)} is a basis for T p M 2 reg and {e 3 (u), . . ., e 2+k (u)} is a basis for N p M 2 reg at p = f (u).

Figure 5 .
Figure 5. Curvature ellipses on the Roman Steiner surface.

Figure 6 .
Figure 6.Curvature ellipses on the elliptic region.
∂x i