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Papers by Carmen Romero Fuster

Homology, Homotopy and Applications

Osaka Journal of Mathematics, 2008

Consider a Riemannian vector bundle of rank 1 defined by a norm al vector field on a surfaceM in ... more Consider a Riemannian vector bundle of rank 1 defined by a norm al vector field on a surfaceM in R4. Let II be the second fundamental form with respect to which determines a configuration of lines of curvature. In th is article, we obtain conditions on to isometrically immerse the surface M with II as a second fundamental form intoR3. Geometric restrictions onM are determined by these conditions. As a consequence, we analyze the extension of Lo ewner’s conjecture, on the index of umbilic points of surfaces in R3, to special configurations on surfaces in R4.

Matemática Contemporânea, 1993

Differential Geometry from a Singularity Theory Viewpoint, 2015

Proceedings of the Steklov Institute of Mathematics, 2009

ABSTRACT With any stable map from a 3-manifold to ℝ3, we associate a graph with weights in its ve... more ABSTRACT With any stable map from a 3-manifold to ℝ3, we associate a graph with weights in its vertices and edges. These graphs are A-invariants from a global viewpoint. We study their properties and show that any tree with zero weights in its vertices and aleatory weights in its edges can be the graph of a stable map from S 3 to ℝ3.

Differential Geometry from a Singularity Theory Viewpoint, 2015

Singularity Theory - Dedicated to Jean-Paul Brasselet on His 60th Birthday - Proceedings of the 2005 Marseille Singularity School and Conference, 2007

Bulletin of the Brazilian Mathematical Society, New Series, 2020

We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into R... more We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into R 4) by means of the simultaneous analysis of the generic singularities of height and squared distanced functions on the flag composed by the 3-manifold, the surface of double points and the crosscaps curve at any point of this curve.

Geometriae Dedicata, 1995

We study the geometry of the surfaces embedded in ~4 through their generic contacts with hyperpla... more We study the geometry of the surfaces embedded in ~4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any genetic convexly embedded 2-sphere in ~4 has inflection points.

We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing n... more We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplane and hyperquadrics in order to get some global informations on them. As a consequence we obtain new versions of Caratèodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings therorems for closed spacelike curves in 3-dimensional Minkowski space.

There are many tools for studying local dynamics. An important problem is how this information ca... more There are many tools for studying local dynamics. An important problem is how this information can be used to obtain global information. We present examples for which local stability does not carry on globally. To this purpose we construct, for any natural n ≥ 2, planar maps whose symmetry group is Zn having a local attractor that is not a global attractor. The construction starts from an example with symmetry group Z 4. We show that although this example has codimension 3 as a Z 4-symmetric map-germ, its relevant dynamic properties are shared by two 1-parameter families in its universal unfolding. The same construction can be applied to obtain examples that are also dissipative. The symmetry of these maps forces them to have rational rotation numbers.

arXiv: Algebraic Topology, 2015

We study first order local invariants of Vassiliev type for Lagrangian immersions with generic pl... more We study first order local invariants of Vassiliev type for Lagrangian immersions with generic planar caustics. For this we produce some examples of 2-parameter families of Lagrangian maps and study their bifurcation diagrams.

Journal of Singularities, 2020

There are two important classes of surfaces in the hyperbolic space. One of class consists of ext... more There are two important classes of surfaces in the hyperbolic space. One of class consists of extrinsic flat surfaces, which is an analogous notion to developable surfaces in the Euclidean space. Another class consists of horo-flat surfaces, which are given by one-parameter families of horocycles. We use the Legendrian dualities between hyperbolic space, de Sitter space and the lightcone in the Lorentz-Minkowski 4-space in order to study the geometry of flat surfaces defined along the singular set of a cuspidal edge in the hyperbolic space. Such flat surfaces can be considered as flat approximations of the cuspidal edge. We investigate the geometrical properties of a cuspidal edge in terms of the special properties of its flat approximations

Transactions of the American Mathematical Society, 2000

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally... more We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

Geometriae Dedicata, 1997

Convexity may imply points of vanishing torsion as the spatial 4-vertex theorem shows. We state h... more Convexity may imply points of vanishing torsion as the spatial 4-vertex theorem shows. We state here that for a simple closed curve to have nowhere vanishing torsion, it must violate convexity hiding at least twice inside its convex hull. Both the ‘4-vertex’ and Lsquo;hiding-twice’ results are generalized by obtaining a relation between the number of vanishing torsion points (vertices) of

Homology, Homotopy and Applications

Osaka Journal of Mathematics, 2008

Consider a Riemannian vector bundle of rank 1 defined by a norm al vector field on a surfaceM in ... more Consider a Riemannian vector bundle of rank 1 defined by a norm al vector field on a surfaceM in R4. Let II be the second fundamental form with respect to which determines a configuration of lines of curvature. In th is article, we obtain conditions on to isometrically immerse the surface M with II as a second fundamental form intoR3. Geometric restrictions onM are determined by these conditions. As a consequence, we analyze the extension of Lo ewner’s conjecture, on the index of umbilic points of surfaces in R3, to special configurations on surfaces in R4.

Matemática Contemporânea, 1993

Differential Geometry from a Singularity Theory Viewpoint, 2015

Proceedings of the Steklov Institute of Mathematics, 2009

ABSTRACT With any stable map from a 3-manifold to ℝ3, we associate a graph with weights in its ve... more ABSTRACT With any stable map from a 3-manifold to ℝ3, we associate a graph with weights in its vertices and edges. These graphs are A-invariants from a global viewpoint. We study their properties and show that any tree with zero weights in its vertices and aleatory weights in its edges can be the graph of a stable map from S 3 to ℝ3.

Differential Geometry from a Singularity Theory Viewpoint, 2015

Singularity Theory - Dedicated to Jean-Paul Brasselet on His 60th Birthday - Proceedings of the 2005 Marseille Singularity School and Conference, 2007

Bulletin of the Brazilian Mathematical Society, New Series, 2020

We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into R... more We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into R 4) by means of the simultaneous analysis of the generic singularities of height and squared distanced functions on the flag composed by the 3-manifold, the surface of double points and the crosscaps curve at any point of this curve.

Geometriae Dedicata, 1995

We study the geometry of the surfaces embedded in ~4 through their generic contacts with hyperpla... more We study the geometry of the surfaces embedded in ~4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any genetic convexly embedded 2-sphere in ~4 has inflection points.

We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing n... more We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplane and hyperquadrics in order to get some global informations on them. As a consequence we obtain new versions of Caratèodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings therorems for closed spacelike curves in 3-dimensional Minkowski space.

There are many tools for studying local dynamics. An important problem is how this information ca... more There are many tools for studying local dynamics. An important problem is how this information can be used to obtain global information. We present examples for which local stability does not carry on globally. To this purpose we construct, for any natural n ≥ 2, planar maps whose symmetry group is Zn having a local attractor that is not a global attractor. The construction starts from an example with symmetry group Z 4. We show that although this example has codimension 3 as a Z 4-symmetric map-germ, its relevant dynamic properties are shared by two 1-parameter families in its universal unfolding. The same construction can be applied to obtain examples that are also dissipative. The symmetry of these maps forces them to have rational rotation numbers.

arXiv: Algebraic Topology, 2015

We study first order local invariants of Vassiliev type for Lagrangian immersions with generic pl... more We study first order local invariants of Vassiliev type for Lagrangian immersions with generic planar caustics. For this we produce some examples of 2-parameter families of Lagrangian maps and study their bifurcation diagrams.

Journal of Singularities, 2020

There are two important classes of surfaces in the hyperbolic space. One of class consists of ext... more There are two important classes of surfaces in the hyperbolic space. One of class consists of extrinsic flat surfaces, which is an analogous notion to developable surfaces in the Euclidean space. Another class consists of horo-flat surfaces, which are given by one-parameter families of horocycles. We use the Legendrian dualities between hyperbolic space, de Sitter space and the lightcone in the Lorentz-Minkowski 4-space in order to study the geometry of flat surfaces defined along the singular set of a cuspidal edge in the hyperbolic space. Such flat surfaces can be considered as flat approximations of the cuspidal edge. We investigate the geometrical properties of a cuspidal edge in terms of the special properties of its flat approximations

Transactions of the American Mathematical Society, 2000

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally... more We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

Geometriae Dedicata, 1997

Convexity may imply points of vanishing torsion as the spatial 4-vertex theorem shows. We state h... more Convexity may imply points of vanishing torsion as the spatial 4-vertex theorem shows. We state here that for a simple closed curve to have nowhere vanishing torsion, it must violate convexity hiding at least twice inside its convex hull. Both the ‘4-vertex’ and Lsquo;hiding-twice’ results are generalized by obtaining a relation between the number of vanishing torsion points (vertices) of