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The purpose of this paper is to investigate |$(+1)$| -shifted Poisson structures in the context of differential geometry. The relevant notion is that of |$(+1)$| -shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of |$(+1)$| -shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (1) We introduce a |${\mathbb{Z}}$| -graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack |${\mathfrak{X}}$|⁠. It turns out that |$(+1)$| -shifted Poisson structures on |${\mathfrak{X}}$| correspond exactly to elements of the Maurer–Cartan moduli set of the corresponding dgla. (2) We introduce the notion of the tangent complex |$T_{\mathfrak{X}}$| and the cotangent complex |$L_{\mathfrak{X}}$| of a differentiable stack |${\mathfrak{X}}$| in terms of any Lie groupoid |$\Gamma{\rightrightarrows } M$| representing |${\mathfrak{X}}$|⁠. They correspond to a homotopy class of 2-term homotopy |$\Gamma$| -modules |$A[1]\rightarrow TM$| and |$T^{\vee } M\rightarrow A^{\vee }[-1]$|⁠ , respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a |$(+1)$| -shifted Poisson structure on a differentiable stack |${\mathfrak{X}}$| defines a morphism |${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$|⁠. [ABSTRACT FROM AUTHOR]

Convex integration and the holonomic approximation theorem are two well-known pillars of flexibility in differential topology and geometry. They may each seem to have their own flavor and scope. The goal of this paper is to bring some new perspective on this topic. We explain how to prove the holonomic approximation theorem for first-order jets using convex integration. More precisely, we first prove that this theorem can easily be reduced to proving flexibility of some specific relation. Then we prove this relation is open and ample, hence its flexibility follows from off-the-shelf convex integration. [ABSTRACT FROM AUTHOR]

We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is related to the differential geometry of planar sections of the surface parallel to and close to the tangent planes, and to the symmetry sets of isophote curves, that is level sets of intensity in a 2-dimensional image. We investigate also the relationship of the vertex curve with the parabolic and flecnodal curves, and the evolution of the vertex curve in a generic 1-parameter family of smooth surfaces. [ABSTRACT FROM AUTHOR]

We introduce two canonical discretizations of nets on generic surfaces, which consist of an one-parameter family of geodesics and its associated family of conjugate lines. The two types of "discrete conjugate semi-geodesic nets" constitute discrete focal nets of circular nets, which mimics the classical connection between surfaces parametrized in terms of curvature coordinates and their focal surfaces on which one corresponding family of coordinate lines are geodesics. The discrete surfaces constructed in this manner are termed circular-geodesic and conical-geodesic nets, respectively, and may be characterized by compact angle conditions. Geometrically, circular-geodesic nets are constructed via normal lines of circular nets, while conical-geodesic nets inherit their name from their intimately related conical nets, which also discretize curvature line parametrized surfaces. We establish a variety of properties of these discrete nets, including their algebraic and geometric 3D consistency, with the latter playing an important role in (integrable) discrete differential geometry. [ABSTRACT FROM AUTHOR]