De Rham Decomposition Research Articles

De Rham Decomposition Theorem for Strongly Convex Kähler–Berwald Manifolds

De Rham Decomposition Theorem for Strongly Convex Kähler–Berwald Manifolds

De Rham Decomposition for Riemannian Manifolds with Boundary

In this paper, we extend the classical de Rham decomposition theorem to the case of Riemannian manifolds with boundary by using the trick of the development of curves.

Intrinsic holonomy and curved cosets of Cartan geometries

We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. To do this, we intrinsically define a holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries.

A de Rham decomposition type theorem for contact sub-Riemannian manifolds

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field xi is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point qin M such that the holonomy group Psi (q) acts reducibly on H(q) yielding a decomposition H(q) = H_1(q)oplus cdots oplus H_m(q) into Psi (q)-irreducible factors. Using parallel transport we obtain the decomposition H = H_1oplus cdots oplus H_m of H into sub-distributions H_i. Unlike the Riemannian case, the distributions H_i are not integrable, however they induce integrable distributions Delta _i on M/xi , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that T(U/xi )=Delta _1oplus cdots oplus Delta _m, and the latter decomposition of T(U/xi ) induces the decomposition of U/xi into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

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 F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of 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. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

Elliptic isometries of the manifold of positive definite real matrices with the trace metric

We study the differential-geometric properties of the loci of fixed points of the elliptic isometries of the manifold of positive definite real matrices with the trace metric. We also give an explicit description of such loci and in particular we find their De Rham decomposition.

Killing Forms on 2-Step Nilmanifolds

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, $k=2$ or $k=3$, we show that the space of Killing $k$-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.

Some results on [formula omitted]-almost Ricci solitons

The purpose of this paper is to investigate some equations of structure for h-almost Ricci soliton which are a natural generalization for almost Ricci solitons. As a result we obtain that a compact nontrivial h-almost Ricci soliton is isometric to a Euclidean sphere with some conditions using the Hodge–de Rham decomposition theorem. Moreover, we acquire an integral formula for the compact h-almost Ricci solitons.

Similarity structures and de Rham decomposition

A similarity structure on a connected manifold M is a Riemannian metric on its universal cover such that the fundamental group of M acts by similarities. If the manifold M is compact, we show that the universal cover admits a de Rham decomposition with at most two factors, one of which is Euclidean. Very recently, after Belgun and Moroianu conjectured that the number of factors was at most one, Matveev and Nikolayevsky found an example with two factors. When the non-flat factor has dimension 2, we give a complete classification of the examples with two factors. In greater dimensions, we make the first steps towards such a classification by showing that M is a fibration (with singularities) by flat Riemannian manifolds; up to a finite covering of M, we may assume that these manifolds are flat tori. We also prove a version of the de Rham decomposition theorem for the universal covers of manifolds with locally metric connections. During the proof, we define a notion of transverse (not necessarily flat) similarity structure on foliations, and show that foliations endowed with such a structure are either transversally flat or transversally Riemannian. None of these results assumes analyticity.

Reducibility in Sasakian geometry

The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham decomposition theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of cone reducible and consider S 3 S^3 bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on S 3 S^3 bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits.

A splitting theorem for spaces of Busemann non-positive curvature

In this paper we introduce a new tool for decomposing Busemann non-positively curved (BNPC) spaces as products, and use it to extend several important results previously known to hold in specific cases like CAT(0) spaces. These results include a product decomposition theorem, a de Rham decomposition theorem, and a splitting theorem for actions of product groups on certain BNPC spaces. We study the Clifford isometries of BNPC spaces and show that they always form Abelian groups, answering a question raised by Gelander, Karlsson, and Margulis. In the smooth case of BNPC Finsler manifolds, we show that the fundamental groups have the duality property and generalize a splitting theorem previously known in the Riemannian case.

On the Controllability of the Rolling Problem onto the Hyperbolic $n$-space

In the present paper, we study the controllability of the control system associated with rolling without slipping or spinning of a Riemannian manifold $(M,g)$ onto the hyperbolic $n$-space ${\mathbb H}^n$. Our main result states that the system is completely controllable if and only if $(M,g)$ is not isometric to a warped product of a special form, in analogy to the classical de Rham decomposition theorem for Riemannian manifolds. The proof is based on the observations that the controllability issue in this case reduces to determining whether $(M,g)$ admits a reducible action of a hyperbolic analog of the holonomy group and a well-known fact about connected subgroups of ${O}(n,1)$ acting irreducibly on the Lorentzian space ${\mathbb R}^{n,1}$.

On the de Rham–Wu decomposition for Riemannian and Lorentzian manifolds

It is explained how to find the de Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Riemannian or Lorentzian manifold.

A parametrized de Rham decomposition theorem for CR manifolds

A parametrized de Rham decomposition theorem for CR manifolds

Decomposable conformal holonomy in Riemannian signature

AbstractVia Cartan and tractor calculus there is an invariant notion of conformal holonomy. Similar to the deRham decomposition theorem of Riemannian geometry, there is a geometric decomposition theorem for conformal holonomy as well. This was established in 3 and 15,17. However, in general, conformal manifolds with decomposable holonomy exhibit singularities. In this paper we solve the singularity case, which is based on the Sl‐doubling construction of 20.

The Hodge–de Rham decomposition theorem and an application to a partial differential equation

The Hodge–de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.

The De Rham Decomposition Theorem For Metric Spaces

We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension.

De Rham decomposition of netted manifolds

By definition a net on a manifold is a family of complementary foliations. Nets and net morphisms between netted manifolds allow to develop basic tools for the decomposability of netted manifolds and of differentiable maps between them. In this way new generalizations of de Rham’s decomposition theorem are obtained.

Riemann-Cartan-Weyl quantum geometry. II Cartan stochastic copying method, Fokker-Planck operator and Maxwell-de Rham equations

We reintroduce the Riemann-Cartan-Weyl (RCW) spacetime geometries of quantum mechanics [Rapoport (1996),Int. J. Theor. Phys.35(2)] in two novel ways: first, through the covariant formulation of the Fokker-Planck operator of the quantum motions defined by these geometries, and second, by stochastic extension of Cartan's development method. The latter is a gauge-theoretic formulation of nonlinear diffusions in spacetime in terms of the stochastic differential geometry associated to the RCW geometries with Weyl torsion. The Weyl torsion plays the fundamental role of describing the first moment (incorporating also the fluctuations due to the second moment) of the stochastic diffusion processes. In this article we present the most general expression of the Weyl torsion one-form given in terms of its de Rham decomposition into the exact component associated with the 0-spin field ϕ and two electromagnetic potentials, one the codifferential of a 2-form and the other a harmonic 1-form. We thus give an original description of the Maxwell theory and its relation to torsion. We associate these electromagnetic potentials with the irreversibility of the diffusions. In an Appendix, we give a self-contained presentation of the theory of diffusions on manifolds and the stochastic calculi as a basis for the Cartan stochastic copying method.

On the product of affine manifolds and the generalized de Rham splitting theorem

We state some theorems about the product-connection of affine connections and prove (with the aid of them) the deRham decomposition theorem for arbitrary affine connections, satisfying a certain curvature condition. There are several versions of deRham theorem (or the proof of it). The story began with the classical result of G. deRham [5], which paper dealt with the positive definite case. Then Kashiwabara showed, that if ~7 is a torsion free aftine connection on M and TpM = Tip 9 T2p, where the 7~p's are invariant subspaces of the holonomy group with induced invariant distributions T1, T2, and if locally ~7 is the product of the affine connections ~71 and ~72 induced on the integral manifolds of 7'1, T2 by ~7, and (moreover) if M is simply connected, then the statement is true globally as well, i.e. M = M1 x M2 and ~7 = ~71 x ~72, where M I ( M z ) are the maximal integral manifolds of 7'1(7'2). Then in 1964 Wu [6], [7] extended the theorem to indefinite metrics. In 1972 Maltz [4] found an extension to arbitrary metric connections (which can have a not vanishing torsion hut this proof is heavily built on the fact, that the connection is a metric one. (See Maltz [4] p. 173). In this article we want to show, that Kashiwabara's local premise holds for every torsion-free connection satisfying certain curvature condition. (It seems unlikely, that we could abandon the vanishing of the torsion, as in Maltz [4], because otherwise we could not have the induced connections ~71, ~72 on the manifolds M1, M2.) So our results is a generalization of the theorems of deRham [5], (or KobayashiNomizu [3]) and Wu [6], [7], but not a generalization of Kashiwabara [2] or Maltz [4]. Moreover we give modifications of the globalization-part of the proof in KobayashiNomizu [3]. Although we could globalize with Kashiwabara's method, or equally well with that of Maltz, we have two reasons not to do so: the proof in [3] is the most wellknown, and the modifications are quite simple. We use the standard notations of Kobayashi-Nomizu [3] all over this article, expect for denoting the derivative of each map by a lower asterisk. If M = M1 •215 is a product of k manifolds, then we denote qi "7ri(q) for q E M . (lri : M1 x_• Mk ---* Mi is the i 'h projection.)