Mixed Curvature Research Articles

The curvature entropy inequalities of convex bodies

Abstract There are many entropy inequalities in geometry, some of them can be seen as the Minkowski inequalities in the form of entropy, which play important roles in convex geometry. In this article, let P P and Q Q be convex bodies, we introduce the mixed curvature entropy M f ( P , Q ) {M}_{f}\left(P,Q) and the curvature entropy E f ( P , Q ) {E}_{f}\left(P,Q) with respect to a continuous function f f . Moreover, we establish the log \log -Minkowski inequalities of the mixed curvature entropy and the curvature entropy for two classes of three-dimensional convex bodies.

On isometric immersions of almost k-product manifolds

A Riemannian manifold endowed with k≥2 complementary pairwise orthogonal distributions is called a Riemannian almost k-product manifold. In the article, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost k-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called the mixed curvature is also related to the extrinsic geometry of a Riemannian almost k-product manifold. Our inequality also contains mixed scalar curvature type invariants related to B.-Y Chen's δ-invariants. Applications are given for isometric immersions of multiply twisted and warped products (we improve some known inequalities by replacing the sectional curvature with our invariant) and to problems of non-immersion and non-existence of compact leaves of foliated submanifolds.

Geometric Inequalities for a Submanifold Equipped with Distributions

The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions.

Integral Formulas for Almost Product Manifolds and Foliations

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with k=2,3 distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds.

Kähler manifolds and mixed curvature

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a “convex combination” of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100–1126] concerning manifolds with negative k k -Ricci curvature and generalize a result of Wu-Yau [Comm. Anal. Geom. 24 (2016), pp. 901–912] and Diverio-Trapani [J. Differential Geom. 111 (2019), pp. 303–314] to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive.

Dynamical Splitting of Spot-producing Magnetic Rings in a Nonlinear Shallow-water Model

Abstract We explore the fundamental physics of narrow toroidal rings during their nonlinear magnetohydrodynamic evolution at tachocline depths. Using a shallow-water model, we simulate the nonlinear evolution of spot-producing toroidal rings of 6° latitudinal width and a peak field of 15 kG. We find that the rings split; the split time depends on the latitude of each ring. Ring splitting occurs fastest, within a few weeks, at latitudes 20°–25°. Rossby waves work as perturbations to drive the instability of spot-producing toroidal rings; the ring split is caused by the “mixed stress” or cross-correlations of perturbation velocities and magnetic fields, which carry magnetic energy and flux from the ring peak to its shoulders, leading to the ring split. The two split rings migrate away from each other, the high-latitude counterpart slipping poleward faster due to migrating mixed stress and magnetic curvature stress. Broader toroidal bands do not split. Much stronger rings, despite being narrow, do not split due to rigidity from stronger magnetic fields within the ring. Magnetogram analysis indicates the emergence of active regions sometimes at the same longitudes but separated in latitude by 20° or more, which could be evidence of active regions emerging from split rings, which consistently contribute to observed high-latitude excursions of butterfly wings during the ascending, peak, and descending phases of a solar cycle. Observational studies in the future can determine how often new spots are found at higher latitudes than their lower-latitude counterparts and how the combinations influence solar eruptions and space weather events.

OHAM analysis of Newtonian heating in mixed convective flow of CNTs over a stretched cylinder

This work reports mathematical modeling and analysis of mixed convection flow of nanofluids with stagnation point by a stretching cylinder. Carbon nanotubes are dispersed as nanomaterial in the baseliquid of water. Heat transfer characteristics are explored through Newtonian heating. Transformation method is employed in order to obtain ODEs. Furthermore series solutions are developed. Variations in flow, temperature, skin friction and local Nusselt number under influential variables have been plotted. It is elaborated that fluid velocity intensifies with rise in curvature parameter, conjugate parameter, mixed convection parameter, nanoparticle volume fraction and velocity ratio parameter. Similarly decay in temperature of the fluid is analyzed for higher velocity ratio parameter while it enhances with increment in nanoparticle volume fraction, mixed convection parameter, conjugate parameter and curvature parameter. Skin friction can be minimized via higher estimations of mixed convection parameter, velocity ratio parameter and nanoparticle volume fraction. Heat transfer rate or cooling process (Nusselt number) is regulated through higher conjugate and curvature parameters. During comparative analysis of SWCNTs with MWCNTs, better performance is shown by SWCNTs. Skin friction coefficient in limiting sense is also compared with previously published articles.

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds, II

We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange equations (which in the case of space-time are analogous to those in Einstein–Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler–Lagrange equations of the mixed Einstein–Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections.

Mixed curvature almost flat manifolds

We prove a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds.

On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

A Riemannian manifold endowed with k>2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for k=2) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections.

An Integral Formula for a Riemannian Manifold with $k>2$ Singular Distributions

Mathematicians have shown interest in manifolds endowed with several distributions, e.g., webs composed of different regular foliations and multiply warped products, as well as distributions having variable dimensions (e.g., singular Riemannian foliations). In this paper, we extend our previous study of the mixed scalar curvature of two orthogonal singular distributions for the case of $k>2$ singular (or regular) pairwise orthogonal distributions, prove an integral formula with this kind of curvature, and illustrate it by characterizing autoparallel singular distributions.

The Isotropic Cosserat Shell Model Including Terms up to $O(h^{5})$. Part II: Existence of Minimizers

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including $O(h^{5})$ terms. The form of the energy allows us to show the coercivity for terms up to order $O(h^{5})$ and the convexity of the energy. Secondly, we consider only that part of the energy including $O(h^{3})$ terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including $O(h^{5})$ the conditions on the thickness $h$ are those considered in the modelling process and they are independent of the constitutive parameter, in the $O(h^{3})$ -case the coercivity is proven under some more restrictive conditions on the thickness $h$ .

An Integral Formula for Singular Distributions

The work is devoted to the differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. We define such distributions as images of the tangent bundle with smooth endomorphisms. We introduce an operator of the divergence type and prove a new divergence theorem. Then we derive a Codazzi type equation for a pair of transverse singular distributions, generalizing the known one for an almost product structure. Tracing our Codazzi type equation gives an expression of a modified divergence of the sum of mean curvature vectors in terms of invariants of distributions including the modified mixed scalar curvature, which allows us to obtain some splitting results. Applying our divergence type theorem to the above expression with a modified mixed scalar curvature, we obtain an integral formula for a pair of transversal singular distributions; the formula generalizes the well-known one with many applications.

Optimization of entropy generation in motion of magnetite-Fe3O4 nanoparticles due to curved stretching sheet of variable thickness

PurposeInvestigation for convective flow of water-based nanofluid (composed of ferric oxide asnanoparticles) by curved stretching sheet of variable thickness is made. Bejan number andentropy generation analysis is presented in presence of viscous dissipation, mixed convectionand porous medium.Design/methodology/approachIn this paper, by using NDSolve of MATHEMATICA, the nonlinear system of equations is solved. Velocity, temperature, Bejan number and entropy generation for involved dimensionless variables are discussed.FindingsIncrease in velocity is depicted for larger curvature parameter, and opposite trend is witnessed for higher nanoparticle volume concentration. Enhancement in temperature is seen for higher Eckert number while reverse behavior is noticed for larger curvature parameter. Entropy rate increases for variation of curvature parameter, Brinkman number and nanoparticle volume fraction. Bejan number decays for mixed convection and curvature parameters.Originality/valueTo the authors’ knowledge, there exists no study yet which describes flow by curved sheet of variable thickness. Such consideration with nanoparticles seems important task. Thus, the main objective here is to determine entropy generation in ferromagnetic nanofluid flow due to variable thickened curved stretching surface. Additionally, effects of Joule heating, porous medium, mixed convection and viscous dissipation are taken into account.

The Einstein-Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

Homotopy study of magnetohydrodynamic mixed convection nanofluid multiple slip flow and heat transfer from a vertical cylinder with entropy generation

Stimulated by thermal optimization in magnetic materials process engineering, the present investigation investigates theoretically the entropy generation in mixed convection magnetohydrodynamic (MHD) flow of an electrically-conducting nanofluid from a vertical cylinder. The mathematical model includes the effects of viscous dissipation, second order velocity slip and thermal slip, has been considered. The cylindrical partial differential form of the two-component non-homogenous nanofluid model has been transformed into a system of coupled ordinary differential equations by applying similarity transformations. The effects of governing parameters with no-flux nanoparticle concentration have been examined on important quantities of interest. Furthermore, the dimensionless form of the entropy generation number has also been evaluated using homotopy analysis method (HAM). The present analytical results achieve good correlation with numerical results (shooting method). Entropy is found to be an increasing function of second order velocity slip, magnetic field and curvature parameter. Temperature is elevated with increasing curvature parameter and magnetic parameter whereas it is reduced with mixed convection parameter. The flow is accelerated with curvature parameter but decelerated with magnetic parameter. Heat transfer rate (Nusselt number) is enhanced with greater mixed convection parameter, curvature parameter and first order velocity slip parameter but reduced with increasing second order velocity slip parameter. Entropy generation is also increased with magnetic parameter, second order slip velocity parameter, curvature parameter, thermophoresis parameter, buoyancy parameter and Reynolds number whereas it is suppressed with first order velocity slip parameter, Brownian motion parameter and thermal slip parameter.

The weighted mixed curvature of a foliated manifold

We introduce the weighted mixed curvature of an almost product (e.g. foliated) Riemannian manifold equipped with a vector field. We define several qth Ricci type curvatures, which interpolate between the weighed sectional and Ricci curvatures. New concepts of the ?mixed-curvature-dimension condition? and ?synthetic dimension of a distribution? allow us to renew the estimate of the diameter of a compact Riemannian foliation and splitting results for almost product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. We also study the Toponogov?s type conjecture on dimension of a totally geodesic foliation with positive weighted mixed sectional curvature.

Mixed sectional-Ricci curvature obstructions on tori

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp’s theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is positive, bounded away from zero. All constants are explicit and depend only on the dimension of the torus.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

Investigation on ethylene glycol Nano fluid flow over a vertical permeable circular cylinder under effect of magnetic field

Abstract In this research, the mixed convection stationary point flow of an incompressible viscous Nano fluid into a vertical permeable circular cylinder along with electric conductivity is analyzed. Ethylene glycol is used as an ordinary liquid, while nanoparticles include copper and silver. The problem has been calculated without the presence of an inductive and electrical magnetic field while taking into account homogeneous and heterogeneous reactions. The strong nonlinear systems calculations are presented using the Numerical Method after non-dimensionalization. Graphical analysis of the effective parameters such as Prandtl number ( Pr ) , permeability parameter ( V w ) , Schmidt number ( Sc ) , magnetic parameter ( M ) , mixed convection parameter ( λ ) and curvature parameter ( γ ) is precisely investigated on the profiles of velocity, concentration and temperature for different nanoparticles. Conclusions indicate that: The thickness of the thermal boundary layer changes more than the thickness of the hydro-dynamic boundary layer for injection and suction. Also, due to the higher thermal conductivity of silver nanoparticles, the temperature increase in these nanoparticles is more than that of copper. In fact, this paper shows that the heat transfer rate increases with the addition of nanoparticles. In addition, the role of the curvature parameter ( γ ) on the concentration profile shows that the concentration profile decreases with the gradual increase of γ .