The Newton transformation and new integral formulae for foliated manifolds

In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature.     

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.     

Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.     

Vrănceanu connections and foliations with bundle-like metrics

We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the notion of a foliated Riemannian manifold of constant transversal Vrănceanu curvature and the notion of a transversal Einstein foliated Riemannian manifold. The geometry of these two classes of manifolds is studied and the relationship between them is determined.     

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively. This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania.     

Rational Homotopy Theory and Nonnegative Curvature

in this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non-negative curved manifolds admit （complete） metrics with non-negative curvature.     

KZ equation,G-opers,quantum Drinfeld–Sokolov reduction,and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 246–259.     

Conformal Riemannian Maps between Riemannian Manifolds,Their Harmonicity and Decomposition Theorems

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.     

A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation

In the setting of a closed Riemannian manifold endowed with a smooth, non-necessarily integrable distribution, we extend a Lichnerowicz type formula which is known to work in the particular case of a transverse bundle associated to a Riemannian foliation. Interesting settings in which non-integrable distributions appear naturally are emphasized. As an application, we consider the distribution as being even dimensional and integrable; we consider also a hermitian line bundle, with a hermitian connection, such that the induced curvature tensor is non-degenerate, and an arbitrary hermitian bundle endowed also with a hermitian connection. Taking the k power of the line bundle and canonically constructing a Spin ^c Dirac operator defined along the leaves of the foliation generated by the distribution, we prove a vanishing result for the half kernel of this operator.     

Minimal Complex Surfaces with Levi–Civita Ricci-flat Metrics

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the(1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics.More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are K¨ahler Calabi–Yau surfaces and Hopf surfaces.     

On the governing equations of motion of nonholonomic systems on Riemannian manifolds

We propose a geometric approach to formulate the governing equations of motion for a class of nonholonomic systems on Riemannian manifolds. We first present a coordinate-free geometric formulation of the D’Alembert–Lagrange equation. Then by explicating this geometric formulation with respect to an arbitrary frame, we obtain the governing equations of motion in generalized form. The governing equations so obtained directly eliminate the dependent variations without using undetermined multipliers. As examples, we apply the formulation to a rigid body and a system with general first-order nonholonomic constraints; we also demonstrate their equivalences to the known results.     

Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Li (J Geom Anal 17:495–511, 2007; Adv Math 223:1924–1957, 2010).     

A nonholonomic Laplace operator

In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defined in an invariant manner and its properties are considered. The method presented for studying it, as well as for the study of other hypoelliptic operators, involves the use of the geometry of nonholonomic manifolds. The nonholonomic metric (Carnot-Carathéodory metric), the Carathéodory measure, and hypoharmonic functions are defined. A theorem on the comparison of the spectra is proved and the connection is established between the bases of eigenfunctions of the ordinary and nonholonomic Laplacians. Conjectures are formulated on the principal term of the spectral asymptotic expansion of the nonholonomic Laplacian, on the structure of the wave fronts, and on the propagation of singularities.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 96–108, 1990.     

Martingale transforms and L p -norm estimates of Riesz transforms on complete Riemannian manifolds

Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder sharp L ^p -inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L ^p -norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞. Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year.     

Toward a Theory of Spaces of Constant Curvature

We construct some examples of three-dimensional metrics of constant curvature defined by the solutions of nonlinear integrable differential equations and their generalizations and study the properties of Riemann extensions of the metrics of constant curvature. We discuss the connection with the theory of normal Riemannian spaces. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 42–54, January, 2006.     

Integral formulae on foliated symmetric spaces

In this article, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan and the second author) for foliations of codimension 1 or unit vector fields and obtain an infinite series of such formulae involving invariants of the Weingarten operator of a unit vector field, of the Jacobi operator in its direction, and their products. We write several such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also covariant derivatives of the Jacobi operator. We work also with foliations of codimension 1 (or vector fields) which admit “good” (in a sense) singularities.     

Cheeger-gromoll type metrics on the (1,1)-tensor bundles

Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.     

Eigenvalue inequalities for the clamped plate problem of $${\mathfrak{L}}_\nu ^2$$ operator

\({{\mathfrak{L}}_{II}}\) operator is introduced by Xin (2015), which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend \({{\mathfrak{L}}_{II}}\) operator to a more general elliptic differential operator \({{\mathfrak{L}}_\nu}\), and investigate the clamped plate problem of bi-\({{\mathfrak{L}}_\nu}\) operator, which is denoted by \({\mathfrak{L}}_\nu ^2\) on the complete Riemannian manifolds. A general formula of eigenvalues for the \({\mathfrak{L}}_\nu ^2\) operator is established. Applying this formula, we estimate the eigenvalues on the Riemannian manifolds. As some further applications, we establish some eigenvalue inequalities for this operator on the translating solitons with respect to the mean curvature flows, submanifolds of the Euclidean spaces, unit spheres and projective spaces. In particular, for the case of translating solitons, all of the eigenvalue inequalities are universal.

     

Variational properties of the Gauss–Bonnet curvatures

The Gauss–Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss–Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss–Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss–Bonnet curvature.