The concept of curvature ellipse at a point of a surface immersed in 4-space has been known since a long time ago [2] and it has proven to be a useful tool in the study of the geometrical properties from both, the local and global viewpoint [1, 3, 4]. Its natural generalization to higher dimensional manifolds is given by the image of a convenient linear projection of a Veronese submanifold of order 2 in the normal space of the manifold at each point [3]. We call it the curvature locus or curvature veronese. The behaviour of the curvature locus of codimension two submanifolds in IRn was investigated in [5], where a especial emphasis was made in the study of submanifolds with flat normal bundle, for which it was shown that the curvature locus becomes a polihedron at each point.
An important feature of the curvature locus for submanifolds of dimension n ≥ 3 immersed in IRn+k, k > 2, is the fact that, for certain pairs (n,k), it may present several topological types. Another interesting aspect is the possible interpretation of its singularities in terms of the behaviour of the principal directions of the normal fields at a given point and hence the possible description of geometrical properties relative to the behaviour of the family of principal configurations on the submanifold in terms of these singularities. Such singularities become especially interesting for some values (n, k). We consider here the case of 3-manifolds immersed in IRn,n ≥ 5 and provide the analytical description of their curvature locus in terms of the coefficients of their fundamental form at each point. We focalize our attention mainly in the study of the pair (3,3) which represents the richest possibility from the viewpoint of the analysis of shapes of the curvature locus of a 3-manifold. In this case, we show that all the possible curvature locus models are the Steiner surfaces of types 1,3,5,6, the ellipsoid, a truncated cone, a convex planar region (that may adopt different geometrical shapes), a segment and a point. We study the connection of their singularities with the behaviour of the set of principal directions and other geometrical properties, such as convexity and existence of quasiumbilic and umbilic directions at a given point.
A relevant question in connection with this study is the investigation of closed 3-manifolds immersed in IR6 whose curvature locus has constant type. A trivial example of this is given by a round 3-sphere, whose curvature locus at each point coincides with the center of the sphere. Other examples may be furnished by convenient products of lower dimensional round spheres. An interesting open question is the existence of a generic embedding of a closed 3-manifold in IR6 with constant shape of its curvature locus.
References:
[1] S.I.R. Costa, S.M. Moraes, M.C. Romero Fuster, Geometric contacts of surfaces immersed in IRn, n ≥ 5. Differential Geom. Appl. 27 (2009), no. 3, 442–454.
[2] K. Kommerell, Riemannsche Flachen in ebenen Raum von vier Dimensionen. Math Ann. 60 (1905), 546–596.
[3] J. A. Little, On singularities of submanifolds of higher dimensional Euclidean spaces. Annali Mat. Pura Appl. 4A 83 (1969) 261–336.
[4] Mochida D. K. H., Romero-Fuster M. C., Ruas M. A. S., The geometry of surfaces in 4-space from a contact viewpoint. Geom. Dedicata 54 (1995), 323–332.
[5] J. J. Nuño-Ballesteros and M. C. Romero-Fuster, Contact properties of codimension 2 subman- ifolds with flat normal bundle. Rev. Mat. Iberoam. 26 (2010), 799–824.