We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=frac{1}{2}pounds _xi varphi$ and $ell := R(cdot,xi)xi$, emphasizing analogies and differences with respect to the contact metric case. Certain identities in
volving $xi$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(mathcal{H}(M), J, theta)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,varphi,xi,eta,g)$ is Sasakian if and only if the corresponding nondegenerate almost $CR$ structure $(mathcal{H}(M), J)$ is integrable and $J$ is parallel along $xi$ with respect to the Bott partial connection.
This article establishes several remarkably simple identities relating certain metric invariants of level curves of real and complex functions. In particular, we relate lengths of level curves to their curvature and to the gradient field of the funct
ion. Some geometric and analytic applications of the results are shown.
We address the problem of determining the hypersurfaces $fcolon M^{n} to mathbb{Q}_s^{n+1}(c)$ with dimension $ngeq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $sin {0, 1}$ for which there exists another
isometric immersion $tilde{f}colon M^{n} to mathbb{Q}^{n+1}_{tilde s}(tilde{c})$ with $tilde{c} eq c$. For $ngeq 4$, we provide a complete solution by extending results for $s=0=tilde s$ by do Carmo and Dajczer and by Dajczer and the second author. Our main results are for the most interesting case $n=3$, and these are new even in the Riemannian case $s=0=tilde s$. In particular, we characterize the solutions that have dimension $n=3$ and three distinct principal curvatures. We show that these are closely related to conformally flat hypersurfaces of $mathbb{Q}_s^{4}(c)$ with three distinct principal curvatures, and we obtain a similar characterization of the latter that improves a theorem by Hertrich-Jeromin. We also derive a Ribaucour transformation for both classes of hypersurfaces, which gives a process to produce a family of new elements of those classes, starting from a given one, in terms of solutions of a linear system of PDEs. This enables us to construct explicit examples of three-dimensional solutions of the problem, as well as new explicit examples of three-dimensional conformally flat hypersurfaces that have three distinct principal curvatures.
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web
foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points of the fir
st kind. Moreover, we study fold singular points of smooth maps.