Riemannian Space Form Research Articles

Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures

The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to non-null framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by $\mathbb{M}_q^n(c)$ the $n$-dimensional space form of index $q=0,1$ and constant curvature $c\neq 0$. We introduce two types of framed Bertrand curves and framed Mannheim curves in $\mathbb{M}_q^3(c)$ by using two different moving frames: the general moving frame and the Frenet-type frame. We investigate geometric properties of these framed Bertrand and framed Mannheim curves in $\mathbb{M}_q^3(c)$ that may have singularities. We then give characterizations for a non-null framed curve to be a framed Bertrand curve or to be a framed Mannheim curve. We show that in special cases these characterizations reduce to the well-known classical formulas: $\lambda \kappa+\mu \tau=1$ for Bertrand curves and $\lambda(\kappa^2+\tau^2)=\kappa$ for Mannheim curves. We provide several examples to support our results, and we visualize these examples by using the Hopf map, the hyperbolic Hopf map, and the spherical projection.

Characterizing linear Weingarten submanifolds in a Riemannian space form via $L$-parabolicity

Characterizing linear Weingarten submanifolds in a Riemannian space form via $L$-parabolicity

First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

New rigidity results for complete LW submanifolds immersed in a Riemannian space form via certain maximum principles

New rigidity results for complete LW submanifolds immersed in a Riemannian space form via certain maximum principles

On some inequalities for metallic Riemannian space forms

The metallic Riemannian space forms are the subject of this article, specifically their slant submanifolds. Several basic inequalities have been derived for these submanifolds and particular conclusions are examined.

Crofton formulas in pseudo-Riemannian space forms

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

Comparison Geometry for an Extension of Ricci Tensor

For a complete Riemannian manifold M with a (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator \({L_A}(u): = div(A\nabla u)\) and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems such as mean curvature comparison theorem, Bishop–Gromov volume comparison theorem, Cheeger–Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applied to some Riemannian hypersurfaces of Riemannian or Lorentzian space forms.

On Immersions of Surfaces into SL(2, ℂ) and Geometric Consequences

Abstract We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including $SL(2,\mathbb{C})$ and the space of geodesics of $\mathbb{H}^3$, and we prove a Gauss–Codazzi theorem in this setting. This approach has some interesting geometric consequences: (1) it provides a model for the transitioning of hypersurfaces among $\mathbb{H}^n$, ${\mathbb{A}}\textrm{d}{\mathbb{S}}^n$, $\textrm{d}{\mathbb{S}}^n$, and ${\mathbb{S}}^n$; (2) it provides an effective tool to construct holomorphic maps to the $\textrm{SO}(n,\mathbb{C})$-character variety, bringing to a simpler proof of the holomorphicity of the complex landslide; and (3) it leads to a correspondence, under certain hypotheses, between complex metrics on a surface (i.e., complex bilinear forms of its complexified tangent bundle) and pairs of projective structures with the same holonomy. Through Bers theorem, we prove a uniformization theorem for complex metrics.

Generalized Wintgen inequality for slant submanifolds in metallic Riemannian space forms

Generalized Wintgen inequality for slant submanifolds in metallic Riemannian space forms

A New Algebraic Inequality and Some Applications in Submanifold Theory

We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms. We derive Chen’s first inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds.

Curves in three dimensional Riemannian space forms

In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve. Using the notion of associate curve we give a new necessary and sufficient condition of which a Frenet curve is a Mannheim curve or Mannheim partner curve in the three dimensional Euclidean space. Then we generalize these conclusions to the curves which lie on the three dimensional Riemannian sphere and the curves which lie in the three dimensional hyperbolic space. We also give the geometric characterizations of these curves. Our methods can be easily used to reveal the properties of the curves on space forms.

An Algebraic Inequality with Applications to Certain Chen Inequalities

We give a simple proof of the Chen inequality for the Chen invariant δ(2,…,2)︸k terms of submanifolds in Riemannian space forms.

Parabolic approaches to curvature equations

We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate Lp-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the Lp-Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli et al. (2007). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures.

Inequalities for generalized normalized $$\delta $$-Casorati curvatures of slant submanifolds in metallic Riemannian space forms

In the present paper, we establish sharp inequalities involving generalized normalized $$\delta $$ -Casorati curvatures for invariant, anti-invariant and slant submanifolds in metallic Riemannian space forms and characterize the submanifolds for which the equality holds.

Optimization on slant submanifolds of golden Riemannian manifolds using generalized normalized $$\delta $$-Casorati curvatures

In the present paper, we derive optimal inequalities involving generalized normalized $$\delta $$ -Casorati curvatures for slant submanifolds in a golden Riemannian space form. We obtain these inequalities by analysing a suitable constrained extrememum problem on submanifold.

(r, k, a, b)-Stability of hypersurfaces in space forms

In a Riemannian space form, we define the (r, k, a, b)-stability concerning closed hypersurfaces, where r and k are entire numbers satisfying the inequality $$0\le k<r\le n-2$$ and a and b are real numbers (at least one nonzero). In this context, when $$b=0$$ , we provide a characterization of the geodesic spheres as critical points of the Jacobi functional associated with the notion of (r, k, a, 0)-stability. Moreover, in the case $$b\not =0$$ , by supposing that a hypersurface $$\Sigma ^n$$ is contained either in an open hemisphere of the Euclidean sphere or in the Euclidean space or in the hyperbolic space, and considering some appropriate restrictions on the constants a and b, we are able to show that $$\Sigma ^n$$ is (r, k, a, b)-stable if, and only if, $$\Sigma ^n$$ is a geodesic sphere.

Upper bounds for Ricci curvatures for submanifolds in Bochner-Kaehler manifolds

Chen established the relationship between the Ricci curvature and the squared norm of meancurvature vector for submanifolds of Riemannian space form with arbitrary codimension knownas Chen-Ricci inequality. Deng improved the inequality for Lagrangian submanifolds in complexspace form by using algebraic technique. In this paper, we establish the same inequalitiesfor different submanifolds of Bochner-Kaehler manifolds. Moreover, we obtain improvedChen-Ricci inequality for Kaehlerian slant submanifolds of Bochner-Kaehler manifolds.

English

We study the first eigenvalue of the Jacobi operator on closed hypersurfaces with constant mean curvature in non-flat Riemannian space forms. Under an appropriate constraint on the totally umbilical tensor of the hypersurfaces and following Melendez’s ideas in J. Melendez (2014) we obtain a new sharp upper bound of the first eigenvalue of the Jacobi operator.

Gap theorems for linear Weingarten surfaces

We deal with complete surfaces in a 3-dimensional Riemannian space form $${\mathbb {Q}}^3_c$$ of constant sectional curvature $$c\in \{-1,0,1\}$$, satisfying a linear Weingarten relation of the type $$K=aH+b$$, where a and b are constant and H and K denote the mean and Gaussian curvatures, respectively. Under suitable constraints on the values of the constants a and b, we obtain gap theorems showing that such a surface must be either totally umbilical or isometric to a hyperbolic cylinder $${\mathbb {H}}^{1}(-\sqrt{1+r^{2}})\times {\mathbb {S}}^{1}(r)$$, when $$c=-1$$, a circular cylinder $${\mathbb {R}}\times {\mathbb {S}}^{1}(r)$$, when $$c=0$$, and a flat torus $${\mathbb {S}}^{1}(\sqrt{1-r^{2}})\times {\mathbb {S}}^{1}(r)$$, when $$c=1$$. Our approach is based on a weak maximum principle for an appropriate Cheng-Yau modified operator.