Principal Bundle Structure of Matrix Manifolds

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r<k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

Analytic Extension of Riemannian Analytic Manifolds and Local Isometries

This article deals with a locally given Riemannian analytic manifold. One of the main tasks is to define its regular analytic extension in order to generalize the notion of completeness. Such extension is studied for metrics whose Lie algebra of all Killing vector fields has no center. The generalization of completeness for an arbitrary metric is given, too. Another task is to analyze the possibility of extending local isometry to isometry of some manifold. It can be done for metrics whose Lie algebra of all Killing vector fields has no center. For such metrics there exists a manifold on which any Killing vector field generates one parameter group of isometries. We prove the following almost necessary condition under which Lie algebra of all Killing vector fields generates a group of isometries on some manifold. Let g be Lie algebra of all Killing vector fields on Riemannian analytic manifold, h⊂g is its stationary subalgebra, z⊂g is its center and [g,g] is commutant. G is Lie group generated by g and is subgroup generated by h⊂g. If h∩(z+[g;g])=h∩[g;g], then H is closed in G.

Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

<pre>Let X = G/H be a semisimple symmetric space of non-compact style. Our purpose is to construct a compact real analytic manifold in which the semisimple symmetric space X = G/H is realized as an open subset and that $G$ acts analytically on it.</pre><pre> By the <span>Cartan</span> decomposition <span>G = KAH,</span> we must <span>compacify</span> the <span>vectorial</span> part <span>A.$</span></pre><pre> In [6], by using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space G/H is realized as an open subset and that G acts analytically on it.</pre><pre>Our construction is a motivation of the <span>Oshima's</span> construction and it is similar to those in N. <span>Shimeno</span>, J. <span>Sekiguchi</span> for <span>semismple</span> symmetric spaces.</pre><pre>In this note, first we will <span>inllustrate</span> the construction via the case of <span>SL (n, </span>R)/SO_e (1, n-1) and then show that the system of invariant differential operators on X = G/H extends analytically on the corresponding compactification. </pre>

Renormalization of vector fields and Diophantine invariant tori

We extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on 𝕋d×ℝℓ. Each Diophantine vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd. Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting 𝒲 to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

Complexification and Hypercomplexification of Manifolds with a Linear Connection

We give a simple interpretation of the adapted complex structure of Lempert–Szöke and Guillemin–Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperkähler metric.

Duality, Quantum Mechanics and (Almost) Complex Manifolds

The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space [Formula: see text], but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a complex structure on [Formula: see text]. When the latter is a complex-analytic manifold admitting just one complex structure, there is a unique quantization whose classical limit is [Formula: see text]. Then the notion of coherence is the same for all observers. However, when [Formula: see text] admits two or more nonbiholomorphic complex structures, there is one different quantization per different complex structure on [Formula: see text]. The lack of analyticity in transforming between nonbiholomorphic complex structures can be interpreted as the loss of quantum-mechanical coherence under the corresponding transformation. Observers using one complex structure perceive as coherent the states that other observers, using a different complex structure, do not perceive as such. This is the notion of a quantum-mechanical duality transformation: the relativity of the notion of a quantum.