The Lichnerowicz theorem on CR manifolds

We obtain a Bochner type formula and an estimate from below on the spectrum of the sublaplacian of a compact strictly pseudoconvex CR manifold.Comment: 21 page

On the boundary behavior of the holomorphic sectional curvature of the Bergman metric

We obtain a conceptually new differential geometric proof of P.F. Klembeck's result that the holomorphic sectional curvature of a strictly pseudoconvex domain approaches (in the boundary limit) the constant sectional curvature of the Bergman metric of the unit ball.Comment: 14 page

On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We study the pseudohermitian sectional curvature of a CR manifold.Comment: 29 page

Jacobi fields of the Tanaka-Webster connection on Sasakian manifolds

We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.Comment: 52 page

Convection of physical quantities of random density

We study the random flow, through a thin cylindrical tube, of a physical quantity of random density, in the presence of random sinks and sources. We model convection in terms of the expectations of the flux and density and solve the initial value problem for the resulting convection equation. We propose a difference scheme for the convection equation, that is both stable and satisfies the Courant–Friedrichs–Lewy test, and estimate the difference between the exact and approximate solutions

On the canonical foliation of an indefinite locally conformal Kähler manifold with a parallel Lee form

We study the semi-Riemannian geometry of the foliation $mathcal F$ of an indefinite locally conformal Kähler (l.c.K.) manifold $M$, given by the Pfaffian equation $omega = 0$, provided that $nabla omega = 0$ and $c = | omega | neq 0$ ($omega$ is the Lee form of $M$). If $M$ is conformally flat then every leaf of $mathcal F$ is shown to be a totally geodesic semi-Riemannian hypersurface in $M$, and a semi-Riemannian space form of sectional curvature $c/4$, carrying an indefinite c-Sasakian structure (in the sense of T. Takahasi). As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem (due to H. Wu) any geodesically complete, conformally flat, indefinite Vaisman manifold of index $2s$, 0 < s < n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold ${mathbb C}H^n_s (lambda )$, 0 < lambda < 1, equipped with the indefinite Boothby metric $g_{s, n}$

On the geometry of coherent state maps

Given a mechanical system whose phase space $\mathfrak{M}$ is equipped with a complex structure J, and a Hermitian line bundle $(E, H) \to \mathfrak{M}$, a coherent state map is an anti-holomorphic embedding \mathscr{K} \to \mathbb{CP} (\mathfrk{M}) built in terms of $(J, H)$, with $\mathfrak{M} = H^0 \Big( \mathfrak{M}, L^2 \mathscr{O} \big( {T^\ast}^{(n,0)} (\mathfrak{M}) \otimes E \big) \Big)$, such that for any pair of classical states $z, \zeta \in \mathfrak{M}$ the number $\langle \mathscr{K} (z), \mathscr{K}(\zeta ) \rangle$ is the transition probability amplitude from the coherent state $\mathscr{K} (z)$ to $\mathscr{K}(\zeta )$. We examine three related questions, as follows: (i) We generalize Lichnerowicz’s theorem (on ± holomorphic maps of finite-dimensional compact Kählerian manifolds) to describe anti-holomorphic maps $\mathscr{K} : \mathfrak{M} \to \mathbb{CP} ( \mathfrak{M})$ as harmonic maps that are absolute minima within their homotopy classes. (ii) If the phase space is a domain $\mathfrak{M} = \Omega \subset \mathbb{C}^n$ and $E \to \Omega$ is a trivial Hermitian line bundle such that $\gamma = H(\sigma_0, \sigma_0) \in AW(\Omega)$ (i.e., $\gamma$ is an admissible weight), we discuss the use of $K_\gamma (z, \zeta )$ [the $\gamma$-weighted Bergman kernel of $\Omega$] vis-a-vis to the calculation of the transition probability amplitudes, focusing on the case where $\Omega = \Omega_n$ is the Siegel domain and $\gamma (z) = \gamma_a (z) = \big( Im (z_n) - |z^\prime|^2 \big)^a$, a > -1. (iii) We study the boundary behavior of a coherent state map $\mathscr{K}: \Omega \to \mathbb{CP}[L^2 H(\Omega_n , \gamma_a )]$

On Schwarzschild's interior solution and perfect fluid star model

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density rho and pressure field p(r) located in a ball r leq r_0. We find a 1-parameter family of time-independent and radially symmetric solutions {(g_a, rho_a, p_a) : -2m &lt; a &lt; 9 kappa M/(4c^2) identifies the “physical” (i.e., such that p_a(r) geq 0 and p_a(r) is bounded in 0 leq r leq r_0) solutions {p_a : a in mathcal{U}_0} for some neighbourhood mathcal{U}_0 subset (-2m , +infty) of a = 0. For every star model {g_a : a_0 &lt; a &lt; a_1}, we compute the volume V(a) of the region r leq r_0 in terms of abelian integrals of the first, second, and third kind in Legendre form