On differential equations classifying a warped product submanifold of cosymplectic space forms

In the present paper, we extend the study of (Ali et al. in J. Inequal. Appl. 2020:241, 2020) by using differential equations (García-Río et al. in J. Differ. Equ. 194(2):287–299, 2003; Pigola et al. in Math. Z. 268:777–790, 2011; Tanno in J. Math. Soc. Jpn. 30(3):509–531, 1978; Tashiro in Trans. Am. Math. Soc. 117:251–275, 1965), and we find some necessary conditions for the base of warped product submanifolds of cosymplectic space form $\widetilde{M}^{2m+1}(\epsilon )$ M ˜ 2 m + 1 ( ϵ ) to be isometric to the Euclidean space $\mathbb{R}^{n}$ R n or a warped product of complete manifold N and Euclidean space $\mathbb{R}$ R .

The base of warped product submanifolds of Sasakian space forms characterized by differential equations

In the present paper, we find some characterization theorems. Under certain pinching conditions on the warping function satisfying some differential equation, we show that the base of warped product submanifolds of a Sasakian space form $\widetilde{M}^{2m+1}(\epsilon )$ M ˜ 2 m + 1 ( ϵ ) is isometric either to a Euclidean space $\mathbb{R}^{n}$ R n or a warped product of a complete manifold N and the Euclidean line $\mathbb{R}$ R .

On the Geometry of Submersions

Subject of present paper is the geometry of foliation defined by submersions on complete Riemannian manifold. It is proven foliation defined by Riemannian submersion on the complete manifold of zero sectional curvature is total geodesic foliation with isometric leaves. Also it is shown level surfaces of metric function are conformally equivalent.

Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We get a result for the minimal case. Additionally, we give examples of a helicoidal surface with a weighted mean curvature and weighted extrinsic curvature.

Path Planning and Replanning for Mobile Robot Navigation on 3D Terrain: An Approach Based on Geodesic

In this paper, mobile robot navigation on a 3D terrain with a single obstacle is addressed. The terrain is modelled as a smooth, complete manifold with well-defined tangent planes and the hazardous region is modelled as an enclosing circle with a hazard grade tuned radius representing the obstacle projected onto the terrain to allow efficient path-obstacle intersection checking. To resolve the intersections along the initial geodesic, by resorting to the geodesic ideas from differential geometry on surfaces and manifolds, we present a geodesic-based planning and replanning algorithm as a new method for obstacle avoidance on a 3D terrain without using boundary following on the obstacle surface. The replanning algorithm generates two new paths, each a composition of two geodesics, connected via critical points whose locations are found to be heavily relying on the exploration of the terrain via directional scanning on the tangent plane at the first intersection point of the initial geodesic with the circle. An advantage of this geodesic path replanning procedure is that traversability of terrain on which the detour path traverses could be explored based on the local Gauss-Bonnet Theorem of the geodesic triangle at the planning stage. A simulation demonstrates the practicality of the analytical geodesic replanning procedure for navigating a constant speed point robot on a 3D hill-like terrain.

Gradient estimates for the heat equation under the Ricci-harmonic map flow

The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold M evolving under the Ricci flow, coupled with the harmonic map flow between M and a second manifold N. We prove Li-Yau type Harnack inequalities and we consider the cases when M is a complete manifold without boundary and when M is compact without boundary.

A 3-DIMENSIONAL SASAKIAN METRIC AS A YAMABE SOLITON

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

A Note on p-Harmonic 1-Forms on Complete Manifolds

In this paper we prove that there is no nontrivial Lq-integrably p-harmonic 1-form on a complete manifold with nonnegatively Ricci curvature (0 < q < ∞).