Search results for " Manifolds"
showing 10 items of 41 documents
New fourfolds from F-theory
2015
In this paper, we apply Borcea-Voisin's construction and give new examples of fourfolds containing a del Pezzo surface of degree six, which admit an elliptic fibration on a smooth threefold. Some of these fourfolds are Calabi-Yau varieties, which are relevant for the $N=1$ compactification of Type IIB string theory known as $F$-Theory. As a by-product, we provide a new example of a Calabi--Yau threefold with Hodge numbers $h^{1,1}=h^{2,1}=10$.
On deformation of Poisson manifolds of hydrodynamic type
2001
We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that the second cohomology group of these manifolds, in the Poisson-Lichnerowicz cohomology, is ``essentially'' trivial. Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous formal deformations of the above manifolds.
Integrable systems, Frobenius manifolds and cohomological field theories
2022
In this dissertation, we study the underlying geometry of integrable systems, in particular tausymmetric bi-Hamiltonian hierarchies of evolutionary PDEs and differential-difference equations.First, we explore the close connection between the realms of integrable systems and algebraic geometry by giving a new proof of the Witten conjecture, which constructs the string taufunction of the Korteweg-de Vries hierarchy via intersection theory of the moduli spaces of stable curves with marked points. This novel proof is based on the geometry of double ramification cycles, tautological classes whose behavior under pullbacks of the forgetful and gluing maps facilitate the computation of intersection…
Approximation of functions over manifolds : A Moving Least-Squares approach
2021
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated…
Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds
2022
We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our main result recovers some nonexistence theorems from the literature, established in the whole Euclidean space.
Minimal unit vector fields
2002
We compute the first variation of the functional that assigns each unit vector field the volume of its image in the unit tangent bundle. It is shown that critical points are exactly those vector fields that determine a minimal immersion. We also find a necessary and sufficient condition that a vector field, defined in an open manifold, must fulfill to be minimal, and obtain a simpler equivalent condition when the vector field is Killing. The condition is fulfilled, in particular, by the characteristic vector field of a Sasakian manifold and by Hopf vector fields on spheres.
Linear invariants of Riemannian almost product manifolds
1982
Using the decomposition of a certain vector space under the action of the structure group of Riemannian almost product manifolds, A. M. Naveira (9) has found thirty-six distinguished classes of these manifolds. In this article, we prove that this decomposition is irreducible by computing a basis of the space of invariant quadratic forms on such a space.
Conjugate unstable manifolds and their underlying geometrized Markov partitions
2000
Abstract Conjugate unstable manifolds of saturated hyperbolic sets of Smale diffeomorphisms are characterized in terms of the combinatorics of their geometrized Markov partitions. As a consequence, the relationship between the local and the global point of view is also made explicit.
Detecting tri‐stability of 3D models with complex attractors via meshfree reconstruction of invariant manifolds of saddle points
2018
In mathematical modeling it is often required the analysis of the vector field topology in order to predict the evolution of the variables involved. When a dynamical system is multi-stable the trajectories approach different stable states, depending on the initialmconditions. The aim of this work is the detection of the invariant manifolds of thesaddle points to analyze the boundaries of the basins of attraction. Once that a sufficient number of separatrix points is found a Moving Least Squares meshfree method is involved to reconstruct the separatrix manifolds. Numerical results are presented to assess the method referring to tri-stable models with complex attractors such as limit cycles o…
FE Calculation Methodology for the Thermodynamic Fatigue Analysis of an Engine Component
2006
Thermo mechanical fatigue problem is treated to define an analysis methodology permitting the strength evaluation by reliability viewpoint. The main difficulty is the lack of both theoretical and experimental information; consequently the problem is treated verifying continually the validity and the limits of the developing solution method. The main task of the activity described in this paper was the development of a numerical methodology, based on FE analyses, for the evaluation of the structural behavior of engine components subjected to thermo mechanical fatigue phenomena. The chosen application was the exhaust manifold of an IC engine; FE analyses were executed following the standard m…