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Daniel V. Tausk

Publications and source records attributed to Daniel V. Tausk.

29 records · Page 2Linked to original sources
arXiv2005

The single-leaf Frobenius Theorem with Applications

Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the formulation of a "one-leaf" analogue of the classical Frobenius integrability theorem in elementary differential geometry. Several applications of the result will be discussed. First, we will give a characterization of symmetric connections arising as Levi-Civita connections of semi-Riemannian metric tensors. Second, we will prove a general version of the classical Cartan-Ambrose-Hicks Theorem giving conditions on the existence of an affine map with prescribed differential at one point between manifolds endowed with connections.

math.DG↗
arXiv2005

Connections compatible with tensors. A characterization of left-invariant Levi--Civita connections in Lie groups

Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi-Civita connection of some semi-Riemannian metric on the group. As a special case, we will consider constant connections in $\R^n$.

math.DG↗
arXiv2002

Spectral Flow, Maslov Index and Bifurcation of semi-Riemannian Geodesics

We give a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see P. M. Fitzpatrick, J. Pejsachowicz, L. Recht, Spectral Flow and Bifurcation of Strongly Indefinite Functionals Part I. General Theory, J. Funct. Anal. 162 (1) (1999), 52-95.) imply that each non degenerate and non null conjugate (or $P$-focal) point along a semi-Riemannian geodesic is a bifurcation point.

math.DG↗
arXiv2001

Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves $γ$ in a differentiable manifold $M$ that are everywhere tangent to a smooth distribution $\mathcal D$ on $M$; such curves are called horizontal. We study the manifold structure of the set $Ω_{P,Q}(M,\mathcal D)$ of horizontal curves that join two submanifolds $P$ and $Q$ of $M$. We consider an action functional $\mathcal L$ defined on $Ω_{P,Q}(M,\mathcal D)$ associated to a time-dependent Lagrangian defined on $\mathcal D$. If the Lagrangian satisfies a suitable hyper-regularity assumption, it is shown how to construct an associated degenerate Hamiltonian $H$ on $TM^*$ using a general notion of {\em Legendre transform} for maps on vector bundles. We prove that the solutions of the Hamilton equations of $H$ are precisely the critical points of $\mathcal L$.

math.OC↗
arXiv2001

An Index Theory for Paths that are Solutions of a Class of Strongly Indefinite Variational Problems

We prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we consider semi-Riemannian manifolds $(M,\mathfrak g)$ admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for $\mathfrak$. In particular we obtain Morse relations for stationary semi-Riemannian manifolds and for the {\em Gödel-type} manifolds.

math.DG↗
arXiv2000

The Morse Index Theorem in semi-Riemannian Geometry

We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the {\em Maslov index} of a semi-Riemannian geodesic, which is a homological invariant and it substitutes the notion of geometric index in Riemannian geometry. Under generic circumstances, the Maslov index of a geodesic is computed as a sort of {\em algebraic count} of the conjugate points along the geodesic. For non positive definite metrics the index of the index form is always infinite; in this paper we prove that the space of all variations of a given geodesic has a {\em natural} splitting into two infinite dimensional subspaces, and the Maslov index is given by the difference of the index and the coindex of the restriction of the index form to these subspaces. In the case of variable endpoints, two suitable correction terms, defined in terms of the endmanifolds, are added to the equality. Using appropriate change of variables, the theory is entirely extended to the more general case of {\em symplectic differential systems}, that can be obtained as linearizations of the Hamilton equations.

math.DG↗
arXiv2000

On the Distribution of Conjugate Points along semi-Riemannian Geodesics

Helfer in [Pacific J. Math. 164/2 (1994), p. 321--350] was the first to produce an example of a spacelike Lorentzian geodesic with a continuum of conjugate points. In this paper we show the following result: given an interval $[a,b]$ of $IR$ and any closed subset $F$ of $IR$ contained in $]a,b]$, then there exists a Lorentzian manifold $(M,g)$ and a spacelike geodesic $γ:[a,b]\to M$ such that $γ(t)$ is conjugate to $γ(a)$ along $γ$ iff $t\in F$.

math.DG↗
arXiv2000

Variational aspects of the geodesic problem in sub-Riemannian geometry

We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds $\mathcal P$ and $\mathcal Q$ of a manifold $\mathcal M$ endowed with a distribution $\mathcal D\subset T\M$. We give a different proof, that holds in a more general context, of a result by Bismut (Large Deviations and the Malliavin Calculus, Birkhauser, 1984) stating that the normal extremizers that are not abnormal are critical points of the sub-Riemannian action functional. We use the Lagrangian multipliers method in a Hilbert manifold setting, which leads to a characterization of the abnormal extremizers (critical points of the endpoint map) as curves where the linear constraint fails to be regular. Finally, we describe a modification of a result by Liu and Sussmann that shows the global distance minimizing property of sufficiently small portions of normal extremizers between a point and a submanifold.

math.DG↗
arXiv2000

Stability of the Focal and Geometric Index in semi-Riemannian Geometry via the Maslov Index

We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic $γ$. For a Riemannian or a non spacelike Lorentzian geodesic, such number is equal to the intersection number (Maslov index) of a continuous curve with a subvariety of codimension one of the Lagrangian Grassmannian of a symplectic space. Such intersection number is proven to be stable in a large variety of circumstances. In the general semi-Riemannian case, under suitable hypotheses this number is equal to an algebraic count of the multiplicities of the conjugate points, and it is related to the spectral properties of a non self-adjoint differential operator. This last relation gives a weak extension of the classical Morse Index Theorem in Riemannian and Lorentzian geometry. In this paper we reprove some results that were incorrectly stated by Helfer in a previous reference; in particular, a counterexample to one of Helfer's results, which is essential for the theory, is given. In the last part of the paper we discuss a general technique for the construction of examples and counterexamples in the index theory for semi-Riemannian metrics, in which some new phenomena appear.

math.DG↗
arXiv1999

Morse Theory for the Travel Time Brachistochrones in Stationary Spacetimes

The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. In this paper we develop a global variational theory for brachistochrones joining an event $p$ and the worldline of an observer $γ$ in a stationary spacetime $M$. More specifically, using the method of Lagrange multipliers, we compute the first and the second variation of the travel time functional, obtaining two variational principles relating the geometry of the brachistochrones with the geometry of geodesics in a suitable Riemannian structure. We present an extension of the classical Morse Theory for Riemannian geodesics to the case of travel time brachistochrones, and we prove a Morse Index Theorem for brachistochrones. Finally, using techniques from Global Analysis, we prove the Morse relations for the travel time functional and we establish some existence and multiplicity results for brachistochrones.

math-ph↗