相关论文: On the contact geometry of nodal sets

We consider local geometry of sub-pseudo-Riemannian structures on contact manifolds. We construct fundamental invariants of the structures and show that the structures give rise to Einstein-Weyl geometries in dimension 3, provided that…

In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $\Delta_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a…

This paper studies distributed-parameter systems on Riemannian manifolds with respect to Stokes-Dirac structures in a language of contact geometry with fiber bundles. For the class where energy functionals are quadratic, it is shown that…

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and…

We consider a 3-dimensional Riemannian manifold with additional structure q. We find a condition that the affine structure q is parallel with respect to the Riamannian connection.We prove the sectional curvatures of three 2-sections formed…

We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analyzed in detail in terms of the relative separations.…

We study combinatorial Laplacians on rectangular subgraphs of $ \epsilon \mathbb{Z}^2 $ that approximate Laplace-Beltrami operators of Riemannian metrics as $ \epsilon \rightarrow 0 $. These laplacians arise as follows: we define the notion…

We classify contact toric 3-manifolds up to contactomorphism, through explicit descriptions, building off of work by Lerman [Lerman03]. As an application, we classify all contact structures on 3-manifolds that can be realised as a concave…

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the…

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of…

We introduce a global Cauchy-Riemann($CR$)-invariant and discuss its behavior on the moduli space of $CR$-structures. We argue that this study is related to the Smale conjecture in 3-topology and the problem of counting complex structures.…

We discuss the geometrical connection between 2D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFTs with monodromies being the discrete subgroups of SL(2,R), the determination…

It has been conjectured by Rovelli that there is a correspondence between the space of link classes of a Riemannian 3-manifold and the space of 3-geometries (on the same manifold). An exact statement of his conjecture will be established…

We establish convergence of spectra of Neumann Laplacian in a thin neighborhood of a branching 2D structure in 3D to the spectrum of an appropriately defined operator on the structure itself. This operator is a 2D analog of the well known…

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step…

We examine the $L^2$-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl Lemma of harmonic analysis, and deduce local pathwise connectedness and local uniform…

This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null…

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a…

Motivated by recent experimental work on multicomponent lipid membranes supported by colloidal scaffolds, we report an exhaustive theoretical investigation of the equilibrium configurations of binary mixtures on curved substrates. Starting…

This paper addresses the problem of friction-free contact between two elastic bodies. We develop an augmented Lagrangian method that provides computational convenience by reformulating the contact problem as a nonlinear variational…