The Gauss-Bonnet theorem for Riemannian polyhedra

1943 ◽  
pp. 299-327
Author(s):  
André Weil
Keyword(s):  
Bonnet Theorem

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

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Kun-Lin Wu ◽  
Ting-Jui Ho ◽  
Sean A. Huang ◽  
Kuo-Hui Lin ◽  
Yueh-Chen Lin ◽  
...  
Keyword(s):  
Mobile Robot ◽  
Robot Navigation ◽  
Tangent Plane ◽  
Intersection Point ◽  
Mobile Robot Navigation ◽  
Complete Manifold ◽  
Geodesic Triangle ◽  
Planning Stage ◽  
3D Terrain ◽  
Bonnet Theorem

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.

105.14 Cubes, cones and the Gauss-Bonnet theorem

2021 ◽  
Vol 105 (562) ◽  
pp. 148-153
Author(s):  
J. N. Ridley
Keyword(s):  
Bonnet Theorem

scholarly journals The Gauss-Bonnet theorem for vector bundles

2006 ◽  
Vol 85 (1-2) ◽  
pp. 15-21 ◽  
Author(s):  
Denis Bell
Keyword(s):  
Vector Bundles ◽  
Bonnet Theorem

scholarly journals Applications of the Gauss–Bonnet theorem to gravitational lensing

2008 ◽  
Vol 25 (23) ◽  
pp. 235009 ◽  
Author(s):  
G W Gibbons ◽  
M C Werner
Keyword(s):  
Gravitational Lensing ◽  
Bonnet Theorem

scholarly journals Weak Deflection Angle of some Classes of non-linear Electrodynamics Black Holes via Gauss-Bonnet Theorem

2020 ◽  
Author(s):  
Hasan El Moumni ◽  
Karima Masmar ◽  
Ali Övgün
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Gravitational Lensing ◽  
Deflection Angle ◽  
Weak Field ◽  
Leading Order ◽  
Plasma Medium ◽  
Non Linear ◽  
The Deflection Angle ◽  
Bonnet Theorem

In this paper, we study the gravitational lensing by some black hole classes within the non-linear electrodynamics in weak field limits. First, we calculate an optical geometry of the non-linear electrodynamics black hole then we use the Gauss-Bonnet theorem for finding deflection angle in weak field limits. The effect of non-linear electrodynamics on the deflection angle in leading order terms is studied. Furthermore, we discuss the effects of the plasma medium on the weak deflection angle.

LAPLACIAN COMPARISON ON COMPLEX FINSLER MANIFOLDS AND ITS APPLICATIONS

2013 ◽  
Vol 24 (05) ◽  
pp. 1350034
Author(s):  
JINXIU XIAO ◽  
CHUNHUI QIU ◽  
QUN HE ◽  
ZHIHUA CHEN
Keyword(s):  
Maximum Principle ◽  
Distance Function ◽  
Finsler Metric ◽  
Finsler Manifolds ◽  
Complex Finsler Metric ◽  
Bonnet Theorem

By defining the Rund Laplacian, we obtain the first and the second holomorphic variation formulas for the strongly pseudoconvex complex Finsler metric. Using the holomorphic variation formulas, we get an estimate for the Levi forms of distance function on complex Finsler manifolds. Further, an estimate for the Rund Laplacians of distance function on strongly pseudoconvex complex Finsler manifolds is obtained. As its applications, we get the Bonnet theorem and maximum principle on complex Finsler manifolds.

The Generalized Gauss-Bonnet Theorem

Tubes ◽  
2004 ◽  
pp. 71-83
Author(s):  
Alfred Gray
Keyword(s):  
Bonnet Theorem
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