In the paper we consider Riemannian surfaces admitting a global expression of the Gauss curvature as the divergence of a vector field. It is equivalent to the existence of a metric linear connection of zero curvature. Such a linear connection $\nabla$ plays an important role in the differential geometry of non-Riemannian surfaces in the sense that the Riemannian quadratic forms can be changed into Minkowski functionals in the tangent planes such that the Minkowskian length of the tangent vectors is invariant under the parallel translation with respect to $\nabla$ (compatibility condition). A smoothly varying family of Minkowski functionals in the tangent planes is called a Finslerian metric function under some regularity conditions. Especially, the existence of a compatible linear connection provides the Finsler surface to be a so-called generalized Berwald surface. It is an alternative of the Riemannian geometry for $\nabla$. Using some general observations and topological obstructions we concentrate on explicit examples. In some representative cases (Euclidean plane, hyperbolic plane etc.) we solve the differential equation of the parallel vector fields to construct a smoothly varying family of Minkowski functionals in the tangent planes such that the Minkowskian length of the tangent vectors is invariant under the parallel translation.
In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.
In this note we give a characterization of taut Riemannian foliations using the transverse divergence. This result turns out to be a convenient tool in the case of some standard examples. Furthermore, we show that a classical tautness result of Haefliger can be obtained in our particular setting as a straightforward consequence. In the final part of the paper we obtain a tautness characterization for transversally oriented foliations with dense leaves and investigate the case of a submanifold of lower dimension which is closed and orthogonal to the leaves.
In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vectors, we propose a novel definition of a canonical divergence and its dual function. We prove that the new divergence allows to recover a given dual structure $(\mathrm{g},\nabla,\nabla^*)$ of {a dually convex set on} a smooth manifold $\mathrm{M}$. Additionally, we show that this d…
We prove some inequalities of Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result for the clamped plate problem. We also prove estimates for lower order eigenvalues that contain some of the estimates from the literature. As an application of our results, we obtain eigenvalues estimates for the bi-drifted Cheng-Yau operator.
The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme $(\mathbb H^1,<,>_L)$, in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of $\mathbb H^1$ when $L\rightarrow\infty$. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in $\mathbb H^1$ as limit of Gauss-Bonnet theorems in $(\mathbb H^1,<,>_L)$ when $L$ goes to infinity. This construction was extended by Wang-Wei in arXiv:1912.00302 to the …
We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and for compact surfaces (which are topologically a torus) we obtain $\int_S K = 0$.
Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space
We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in $\mathbb{H}$ is provided.
We verify if Gausssian curvature of surfaces and normal curvature of curves in surfaces introduced by Diniz-Veloso arXiv:1210.7110 and by Balogh-Tyson-Vecchi arXiv:1604.00180 to prove Gauss-Bonnet theorems in Heisenberg space $\mathbb H^1$ are equal. The authors in arXiv:1604.00180 utilize a limit of Gaussian and normal curvatures defined in the Riemannian approximations scheme $(\mathbb R^3,g_L)$ introduced by Gromov to study sub-Riemannian spaces. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in $\mat…
Let M be a compact Riemannian manifold and E a Riemannian vector bundle on M. We look for hypersurfaces of E with a prescribed vertical Gaussian curvature. In trying to solve this problem fibre-wise, we loose the regularity of the resulting solution. To unsure the smoothness of the solution, we construct it as a radial graph over the unit sphere subbundle of E and prove its existence by solving in this one a nonlinear partial differential equation of Monge-Ampère type.
For a Riemannian $G$-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost harmitian and almost contact metric structures.
In this paper, an optimal inequality involving the delta curvature is exposed. An application of Riemannian submersions dealing meteorology is presented. Some characterizations about the vertical motion and the horizontal divergence are obtained.
We consider translating solitons to flows by positive powers $α$ of the Gaussian curvature -- called $K^α$-flows -- in Riemannian products $M\times\mathbb R.$ We prove that, when $M$ is the Euclidean space $\mathbb R^n,$ the sphere $\mathbb S^n,$ or one of the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m,$ there exist complete rotational translating solitons to $K^α$-flow in $M\times\mathbb R$ for certain values of $α.$
In this note, we prove two Kazdan-Warner type identities involving $v^{(2k)}$, the renormalized volume coefficients of a Riemannian manifold $(M^n,g)$, and $G_{2r}$, the so-called Gauss-Bonnet curvature, and a conformal Killing vector field on $(M^n,g)$. In the case when the Riemannian manifold is locally conformally flat, $v^{(2k)}=(-2)^{-k}σ_k$, $G_{2r}(g)=\frac{4^r(n-r)!r!}{(n-2r)!}σ_r$, and our results reduce to earlier ones established by Viaclovsky and by the second author.
This expository paper contains a detailed introduction to some important works concerning the Gauss-Bonnet-Chern theorem. The study of this theorem has a long history dating back to Gauss's Theorema Egregium (Latin: Remarkable Theorem) and culminated in Chern's groundbreaking work [14] in 1944, which is a deep and wonderful application of Elie Cartan's formalism. The idea and tools in [14] have a great generalization and continue to produce important results till today. In this paper, we give four different proofs of the Gauss-Bonnet-Chern theorem on Riemannian manifolds, namely Chern's simple…
We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hypersurface of the unit sphere $\mathbb{S}^{n+1}$ ($n\geq2$) contained in a geodesic ball of radius $R$ and whose principal curvatures are strictly bigger than $\tan\left( R/2 \right)$, then $M$ is diffeomorphic to $\mathbb{S}^{n}$. Additionally, we show that for any $\varepsilon\…
We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is geodesically convex. The geodesic equation for this metric corresponds to a degenerate elliptic fully nonlinear PDE, and we prove that any two points are connected by a $C^{1,1}$ geodesic. Using this we can define a length space structure on the given conformal class. We present a different approach to the uniformization theorem by studying the negative gradient flow …
On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of noncompact complete preimage and target manifolds is considered. We give conditions for existence and uniqueness of Hermitian-harmonic maps and solutions of the corresponding parabolic system, which observe the non-divergence form of the underlying equations. Numerous examples illustrate the theoretical results and the fundamental difference to harmonic maps.
We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. We also introduce some applications of this curvature identity.
In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts. The first one is a new form of the classical Ahlfors maximum principle whereas the second one is a version for manifolds with boundary of the so called Kelvin-Nevanlinna-Royden criterion of parabolicity. In fact, we will show that the validity of non-compact versions of these tools serve as a characterization of the Neumann parabolicity of the space. The mo…
In this paper, we study the geometry of a connected oriented cmc Riemannian hypersurface $M$ of a semi-Riemannian group $G$ of Lie algebra $\mathfrak g$ and index 0 or 1. If $G$ is Riemannian and $M$ is compact and transversal to an element of $\mathfrak g$, we show that it is a lateral class of a closed embedded Lie subgroup of $G$; we also do this if $G$ is Lorentzian, provided $M$ has sufficiently large mean curvature. If $G$ is Riemannian semisimple and $M$ is compact, we prove that $M$ has degenerate Gauss map and minimal relative nullity at least 1. We also extend the above results to th…
We give a curvature identity derived from the generalized Gauss-Bonnet formula for 4-dimensional compact oriented Riemannian manifolds. We prove that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact. We also provide some applications of the identity.
We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient flow and the scalar curvature of this space based on modular operator are given. We also show the Gauss-Bonnet theorem for a parametrized class of non-diagonal metrics on noncommutative two-torus.
We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle $E$ of even rank over a closed compact orientable manifold $M$. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when $M$ is a Riemannian manifold and $E$ is the tangent bundle of $M$ endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles.
In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold $M\times\mathbb{R}$. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal graphs bounded by two Jordan curves in parallel planes. The key to the proofs is the Weierstrass representation of those surfaces via $\wp-$harmonic mappings. We also prove some Schwarz lemma type results and some Heinz type results for harmonic mappings between geodesic disks in Riemannian surfaces.
Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate-invariant properties of statistical inference. Their connections and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential geometric connection between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, a geometric perspective on the fundam…
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures…
Arnlind, Hoppe and Huisken showed how to express the Gauss and mean curvature of a surface embedded in a Riemannian manifold in terms of Poisson brackets of the embedding coordinates. We generalize these expressions to the pseudo-Riemannian setting and derive explicit formulas for the case of surfaces embedded in $\R^m$ with indefinite metric.
We derive the Chern-Gauss-Bonnet Theorem for manifolds with smooth non-degenerate boundary in the pseudo-Riemannian context from the corresponding result in the Riemannian setting by examining the Euler-Lagrange equations associated to the Pfaffian of a complex "metric" on the tangent space and then applying analytic continuation.
We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $Σ$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a harmonic diffeomorphism from $S$ onto $Σ$. The proof uses the theory of divergence lines to construct minimal graphs. We also generalize a theorem of R. Schoen. Let $g_1$ and $g_2$ be two complete metrics on a orientable surface $S$ with compact boundary and suppose $$\int_{S_r^2}K_{g_2}^-dσ_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all $r>0$. If there is a harmonic diffe…
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R^3. We prove a general local existence result for a large class of Monge-Ampere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth cu…
Motivated by a recent work of X. Chen and M. Zhu (Commun. Math. Stat., 1 (2013) 369-385), we establish a Trudinger-Moser inequality on compact Riemannian surface without boundary. The proof is based on blow-up analysis together with Carleson-Chang's result (Bull. Sci. Math. 110 (1986) 113-127). This inequality is different from the classical one, which is due to L. Fontana (Comment. Math. Helv., 68 (1993) 415-454), since the Gaussian curvature is involved. As an application, we improve Chen-Zhu's result as follows: A modified Liouville energy of conformal Riemannian metric has a uniform lower …
We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus proving the Riemannian positive mass theorem in this case. If the graph has convex horizons, we also prove the Riemannian Penrose inequality by giving a lower bound to the boundary integrals using the Aleksandrov-Fenchel inequality.
The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo's problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss-Bonnet formula in an inequality. This Gauss-Bonnet inequality enables to generalize for Zermelo's problems the E. Hopf theorem on flatness of Riemannian tori without conjugate points.
We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure …
We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate all the previous curvatures. We then examine the curvature term which appears in the classical Weitzenböck formula. We also study the positivity properties of the $p$-curvatures, the second Gauss-Bonnet-Weyl curvature, the Einstein curvature and the isotropic curvature.
The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz-Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more general…
In this paper we consider the Gaussian thermostat ray transform on both closed Riemannian surfaces and compact Riemannian surfaces with boundary. We establish certain results on the injectivity of the thermostat ray transform and the surjectivity of its adjoint.
We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_λ$ over geodesics go to zero at the rate of $O((\logλ)^{-1/2})$ if $λ$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible in the constant curvature case if the curvature is $\ge0$. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius $r\le 1$ are pinched from above by $-δr^N$ …
