In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics:The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R 3 are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in 1540008-1 M. Hasegawa et al.these classes of metrics. Moreover, we prove Gauss-Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds. Lemma 2.2 (Kossowski [7]). Let p be a singular point of dσ 2 . Then the Kossowski pseudo-connection Γ induces a tri-linear mapwhere V j (j = 1, 2, 3) are vector fields of M 2 such that v j = V j (p). 1540008-4Intrinsic properties of surfaces with singularities Proof. Applying (2.1),holds at p, where the fact that V 3 (p) ∈ N p is used to show the last equality. Definition 2.3. A singular point p of the metric dσ 2 is called admissible b ifΓ p in Lemma 2.2 vanishes. If each singular point of dσ 2 is admissible, then dσ 2 is called an admissible metric.We are interested in admissible metrics because of the following fact. Proposition 2.4. Let f : M 2 → R 3 be a C ∞ -map. Then the induced metric dσ 2 (:= df · df ) by f on M 2 is an admissible metric.Proof. Let D be the Levi-Civita connection of the canonical metric of R 3 . Then the Kossowski pseudo-connection of dσ 2 is given by (cf. (2.2))which vanishes if df (Z p ) = 0, proving the assertion.A singular point of the metric dσ 2 is called of rank one if N p is a 1-dimensional subspace of T p M 2 .Definition 2.5. Let dσ 2 be a positive semi-definite metric on M 2 . A local coordinate system (U ; u, v) of M 2 is called adjusted at a singular point p ∈ U if Moreover, if (U ; u, v) is adjusted at each singular point of U , it is called an adapted local coordinate system of M 2 .By a suitable affine transformation in the uv-plane, one can take a local coordinate system which is adjusted at a given rank one singular point p. Lemma 2.6. Let (ξ, η) and (u, v) be two local coordinate systems centered at a rank one singular point p. Suppose that (u, v) is adjusted at p = (0, 0). Then (ξ, η) is also adjusted at p if and only if u η (0, 0) = 0 (2.6) holds. b The admissibility was originally introduced by Kossowski [7]. He called it d( , )-flatness. 1540008-5 M. Hasegawa et al.

Abstract. It is classically known that generic smooth maps of R 2 into R 3 admit only cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap f std (u, v) = (u, uv, v 2 ) has non-trivial isometric deformations with infinite dimensional freedom. Since there are several geometric invariants for cross caps, the existence of isometric deformations suggests that one can ask which invariants of cross caps are intrinsic. In this paper, we show that there are three fundamental intrinsic invariants for cross caps. The existence of extrinsic invariants is also shown.

Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular points on spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces do not admit fold singular points. Moreover, we show that the singular point set of any conjugate CMC surface of a spacelike Delaunay surface with conelike singular points consists of (2, 5)-cuspidal edges.

Abstract. In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with nonvanishing mean curvature and investigate their properties.

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean 3-space. Their first fundamental forms belong to a class of positive semi-definite metrics called 'Kossowski metrics'. A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points (the definition of 'non-parabolic singular point' is stated in the introduction here) can be realized as wave front germs (Kossowski's realization theorem).On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of 'coherent tangent bundle'. Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle.In this paper, we shall explain Kossowski's realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge singularity (resp. swallowtail singularity or cuspidal cross cap singularity). Several applications of these criteria are given. Some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.