Marek Kossowski

Transverse, Type Changing, Pseudo Riemannian Metrics and the Extendability of Geodesics

We study the geodesics and pre-geodesics of a smooth manifold with smooth pseudo riemannian metric which changes bilinear type (i. e. the signature changes) on a hypersurface. We classify all geodesics and pre-geodesics that cross the hypersurface of type change transversely. We then apply these results to the eikonal partial differential equation to find geometric conditions for the local existence or non-existence of smooth, transverse solutions.

The Einstein Equation for Signature Type Changing Spacetimes

We give a local existence and uniqueness theorem for solutions of Einstein’s equations with dust energy momentum tensor in the class of m-dimensional, analytic, transverse, signature type changing spacetimes where the initial conditions are given on the hypersurface of signature type change. We also prove a similar theorem in the case that the energy momentum tensor represents a scalar field.

The intrinsic conformal structure and Gauss map of a light-like hypersurface in Minkowski space

We begin by pointing out two subtleties in the global properties of hypersurfaces in Minkowski space which inherit a uniformly degenerate metric (i.e., the existence of global space-like sections and the notion of an icon; see Appendices 1 and 2). We then construct a Gauss map for such hypersurfaces and an intrinsic invariant. This leads us to results concerning light-like hypersurfaces which parallel known results concerning surfaces in Euclidean space.

principles for hypersurfaces with prescribed principle curvatures and directions

We prove that any compact orientable hypersurface with boundary immersed (resp. embedded) in Euclidean space is regularly homotopic (resp. isotopic) to a hypersurface with principal directions which may have any prescribed homotopy type, and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant. Further we construct regular homotopies (resp. isotopies) which control the principal curvatures and directions of hypersurfaces in a variety of ways. These results, which we prove by holonomic approximation, establish some h-principles in the sense of Gromov, and generalize theorems of Gluck and Pan on embedding and knotting of positively curved surfaces in 3-space.

The Lagrangian Gauss image of a surface in Euclidean 3-space

We describe a correspondence between special nonimmersed surfaces in Euclidean 3-space and exact Lagrangian immersions in the cotangent bundle of the unit sphere. This provides several invariants for such nonimmersive maps: the degree of the Gauss map, the Gauss-Maslov class, and the polarization index. The objectives of this paper are to compare these invariants in the cases where the underlying map immerses or fails to immerse and to describe the extend to which these invariants can be prescribed

PRESCRIBING INVARIANTS FOR INTEGRAL SURFACES IN THE GRASSMANN BUNDLE OF 2-PLANES IN 4-SPACE?

A SMOOTHLY immersed integral surface M in the Grassmann bundle M > G,(R4) > R4 defines a (possibly nonimmersed) surface at 0 s(M) in R4. Moreover, a surface in R4 whose nonimmersive singularities are sufficiently well behaved, lifts to an integral surface in G,(R4). (This lift is defined by assigning a “tangent” 2-plane to each point on the surface.) Thus a compact integral surface in G,(R4) can be thought of as a resolution or lift of a nonimmersed surface in R4 which is well defined up to smooth diffeomorphisms of R4 and M. In this paper we will consider the set of compact orientable rank 1 transverse integral surfaces in G,(R4) [4]. We then define five differential topological invariants for such a surface (Section 2). In Theorem 1 we show that these invariants are independent and can be freely prescribed. As a consequence, any 2-plane bundle can be realized as the pull-back bundle s* (6) over a rank 1 transverse integral surface, where & is the tautological2-plane bundle over G2(R4) (Corollary 2). The proof of Theorem 1 is constructive and builds the desired integral surface by cutting and pasting the local models described in Section 3.2. Integral surfaces occur naturally as (multivalued) solutions to PDE [4]. Since a PDE can be viewed as a constraining submanifold in G2(R4) the invariants of an integral surface solution should be restricted by the PDE. In Section 5 (Theorem 3) we briefly describe such restrictions in the case where the PDE is type-homotopic to a determined linear elliptic or hyperbolic system on the plane (cf. [2, 3, 51).

Fully stratified compact hypersurfaces in Minkowski 4-space

A compact hypersurface in Minkowski space decomposes as a disjoint union of loci where the induced metric is: definite, degenerate, or indefinite. Here we deduce several global properties of the hypersurface from properties of its degenerate loci.

The Lagrangian Gauss image of a compact surface in Minkowski 3-Space

A generic compact surfaceQ in Minkowski 3-Space is naturally stratified by the loci where the orthogonal line bundle is tangent to the next lower stratum,SP ⊂D0 ⊂Q ⊂ M3. To each component inD0 we associate a light-like hypersurface and in turn a Lagrangian loop in the cotangent bundle of the circle. We then establish an inequality relating the Euler characteristic of the indefinite component ofQ with the total Gauß-Maslov index of the associated Lagrangian loops.

A Gauss map and hybrid degree formula for compact hypersurfaces in Minkowski space

A compact oriented hypersurface in four-dimensional Minkowski space contains a locus where the induced metric is degenerate. We show that if certain transversality conditions are satisfied then the degree of the hypersurface Gauss map is determined by the Euler characteristic of the degeneracy locus. This leads to a total curvature inequality with lower bound determind by the Betti numbers of the hypersurface. Equality characterizes particularly simple hypersurfaces.

Global Differential Geometry of 1-Resolvable C∞ Curves in the Plane

A C∞ nonimmersed curve in the Euclidean plane c: S1 → E2 is 1-resolvable if its lift to the orthonormal frame bundle c: S1 → F extends to an immersion. The objective of this paper is to relate the shape of the planar curve with the differential topology of its lift. Specifically, we derive inequalities relating geometric invariants of c with topological invariants of c. The corresponding equalities will identify the simplest 1-resolvable curves.