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Let $\pi :Z\to X$ be a holomorphic submersion from a complex manifold $Z$ onto a 1-convex manifold $X$ with exceptional set $S$ and $a:X\to Z$ a holomorphic section. Let $\phi :X\to [0,\infty )$ be a plurisubharmonic exhaustion function which is strictly plurisubharmonic on $X\backslash S$ with ${\phi }^{-1}(0)=S.$ For every holomorphic vector bundle $E\to Z$ there exists a neighborhood $V$ of $a(U\backslash S)$ for $U={\phi }^{-1}([0,c)),$ conic along $a(S),$ such that $E{|}_V$ can be endowed with Nakano strictly positive Hermitian metric. Let $g:X\to ℂ,$$g^{-1}(0)\supset S$ be a given holomorphic function. There exist finitely many bounded holomorphic vector fields defined on a Stein neighborhood $V$ of $a(\overline{U}\backslash g^{-1}(0)),$ conic along $a(g^{-1}(0))$ with zeroes of arbitrary high order on $a(g^{-1}(0))$ and such that they generate $\mathrm{\text{ker}}D\pi {|}_{a(\overline{U}\backslash g^{-1}(0))}.$ Moreover, there exists a smaller neighborhood $V^{\prime }\subset V$ such that their flows remain in $V^{\prime }$ for sufficiently small times thus generating a local dominating spray.

The study of recently introduced Fedosov supermanifolds¹ is continued. Using normal coordinates, properties of even and odd symplectic supermanifolds endowed with a symmetric connection respecting given sympletic structure are studied.

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and Wolak, allowed us to take a new look at the theory of contact manifolds. In this paper, we continue our study of a new structure of this type, called a weak nearly Sasakian structure, and prove theorems characterizing Sasakian manifolds. Our main result generalizes the theorem by Nicola–Dileo–Yudin (2018) and provides a new criterion for a weak almost-contact metric manifold to be Sasakian.