Riemannian Scalar Curvature Research Articles

Scalar curvatures in almost Hermitian geometry and some applications

On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the Kähler metric, the balanced metric, the locally conformal Kähler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier (2020) and a conjecture by Angella et al. (2018).

Scalar Curvatures of Invariant Almost Hermitian Structures on Generalized Flag Manifolds

In this paper we study invariant almost Hermitian geometry on generalized flag manifolds. We will focus on providing examples of K\"ahler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_{\rm C}$, where $s$ is Riemannian scalar curvature and $s_{\rm C}$ is the Chern scalar curvature.

Scalar curvature, Kodaira dimension and $${{\widehat{A}}}$$-genus

Let (X, g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X, J) is equal to $$-\infty $$ and the canonical bundle $$K_X$$ is not pseudo-effective. We also introduce the complex Yamabe number $$\lambda _c(X)$$ for compact complex manifold X, and show that if $$\lambda _c(X)$$ is greater than 0, then $$\kappa (X)$$ is equal to $$-\infty $$ ; moreover, if X is also spin, then the Hirzebruch A-hat genus $${{\widehat{A}}}(X)$$ is zero.

On the proportionality of Chern and Riemannian scalar curvatures

On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler setting, one can seek to understand extensions of these characterizations to general Hermitian manifolds. This idea has been the subject of much study from the cohomological side, however, the focus here is to address such a question from the perspective of curvature relationships. In particular, on compact manifolds the Kahler condition is characterized by the relationship that the Chern scalar curvature is equal to half the Riemannian scalar curvature. What we study here is the existence, or lack thereof, of non-Kahler Hermitian metrics for which a more general proportionality relationship between these scalar curvatures holds.

Ricci curvatures on Hermitian manifolds

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the ( 1 , 1 ) (1,1) -component of the curvature 2 2 -form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-Kähler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds S 2 n − 1 × S 1 \mathbb {S}^{2n-1}\times \mathbb {S}^1 . We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and non-negative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with non-negative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.

An Equivalence of Scalar Curvatures on Hermitian Manifolds

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu–Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kahler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

Killing spinor equations in dimension 7 and geometry of integrable G2-manifolds

We compute the scalar curvature of seven-dimensional G 2-manifolds admitting a G 2-connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n=7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G 2-structure into a cocalibrated one of pure type W 3.

Path Integral Quantization of the Aharonov–Bohm–Coulomb System in Momentum Space

The Coulomb system with a charge moving in the fields of Ahanorov and Bohm is quantized via path integral in momentum space. Due to the dynamics of the system in momentum space being in curve space, our result not only gives the Green function of this interesting system in momentum space but provides the second example to answer an open problem of quantum dynamics in curved spaces posed by DeWitt in 1957: We find that the physical Hamiltonian in curved spaces does not contain the Riemannian scalar curvature R.

Riemannian and Finslerian geometry in thermodynamics

It is shown how the methods of Riemannian and Finslerian geometry may be used in thermodynamics of equilibrium and nonequilibrium states. In both cases the Riemannian structure on the spaces of thermodynamic parameters is defined by means of the relative information (entropy). Thermodynamic meaning of the Riemannian scalar curvature is then interpreted in terms of stability of the considered systems. For nonequilibrium systems the time derivative of the relative information leads to the Finslerian structure. It is shown how a homogenization procedure of Rund leads to the Finslerian metric of the Kropina type. Three types of the Finslerian curvature tensors connected with the Cartan connection are considered for two-dimensional spaces. In particular, the so-called horizontal curvature is considered in detail. It turns out that in thermodynamic spaces Cartan connection coincides with the Berwald connection. Thermodynamic meaning of the Finslerian scalar curvatures is not clear since they vanish for two-dimensional spaces.

Nonlinear filtering and Riemannian scalar curvature, [formula omitted

In the present paper on high dimensional nonlinear filtering problems and stochastic partial differential equations, the Riemannian scalar curvature R is shown to play a fundamental role. The n-dimensional signal processes, n ≥ 2, and the m-dimensional observations processes, m ≥ 1, are assumed to be homogeneous strong Markov diffusions with independent noise functionals. The curvature R of the covariance of the diffusion signal explicitly enters the c ∞-density of the measure-valued solution of the Zakai equation for the problems considered here. The special class of signals required for our proofs are not necessary for n = 2. The general result under normalized conditions is that estimations are relatively better if R is positive and worse if R is negative. The results are applied to estimation problems arising in modelling starfish/coral interactions, but can readily be applied to chemically mediated plant/herbivore interactions.

Some curvature properties of complex surfaces

In this paper, we are investigating curvature properties of complex two-dimensional Hermitian manifolds, particularly in the compact case. To do this, we start with the remark that the fundamental form of such a manifold is integrable, and we use the analogy with the locally conformal KÄhler manifolds, which follows from this remark. Among the obtained results, we have the following: a compact Hermitian surface for which either the Riemannian curvature tensor satisfies the KÄhler symmetries or the Hermitian curvature tensor satisfies the Riemannian Bianchi identity is KÄhler; a compact Hermitian surface of constant sectional curvature is a flat KÄhler surface; a compact Hermitian surface M with nonnegative nonidentical zero holomorphie Hermitian bisectional curvature has vanishing plurigenera, c1(M) ⩾ 0, and no exceptional curves; a compact Hermitian surface with distinguished metric, and positive integral Riemannian scalar curvature has vanishing plurigenera, etc.