Geometry Of Submanifolds Of Matrix Lie Groups And Their Applications

Lie theory(i.e.the theory of Lie groups,Lie algebras and their applications)is an important branch of modern mathematics,which is represented by matrix Lie groups.As the name implies,matrix Lie group is both Lie group and matrix group.This is a class of Lie groups that appear widely in mathematics and physics,including general linear groups,orthogonal groups,rigid body motion groups,symplectic groups,unitary groups,etc.Among them,the rigid body motion group SE(3)is one of the core concepts of robotics,and the screw algebra used in the mainstream research of robotics is the subalgebra of the Lie algebra of SE(3).In this paper,we mainly study the Riemannian geometry of SE(3),the local theory of curves in it and the geometrical properties of the envelopes of the the multi-parameter families of surfaces generated by its submanifolds acting on segments of plane curves.In addition,the stationary acceleration curve in SE(2)and the ruled surface with nonzero constant mean curvature in 3-dimensional Heisenberg group are also studied.Firstly,we consider S E(3)as a submanifold of the 12-dimensional Euclidean space,and derive the expressions for its structural equation and Riemann curvature tensor.Then we give the Frenet typed formula for the curve in SE(3)and the expressions for the covariant curvature of each order.In addition,we construct some kinds of curves in S E(3)for which the highest order covariant curvature is zero,and discuss the influence of the different behaviors of the rotation component and the transformation component on the curves themselves.Secondly,we study 2-and 3-dimensional submanifolds of SE(3)and their applications in envelope theory.We first study the geometry of 2-and 3-dimensional subalgebras of Lie algebra se(3),and construct the corresponding submanifolds of SE(3)by using exponential map.Then,we study the geometric properties of the envelopes of the multi-parameter families of surfaces generated by curves in different submanifolds acting on segments of plane curves(mainly line segments and arcs).We give the expressions of the Gaussian curvature and the mean curvature of different envelops and draw pictures to describe the enveloping process.Our results show that the final envelope of a given family of surfaces is independent of the order of enveloping process with respect to different parameters.Thirdly,we study the stationary acceleration curve in SE(2).According to the existing conclusion,a curve G(t)in SE(2)is a stationary acceleration curve if and only if V=GXG-1,where V=GG-1 and X is a constant vector.Under this constraint condition,the rotation angle of motion is at most a cubic function of the variable,and the main difficulty lies in solving the translation variable.We give the sufficient and necessary conditions for the curves whose rotation angle are quadratic functions or cubic functions without 1st and 2nd order terms to be stationary acceleration curves.The results obtained are an extension to the existing conclusions.Finally,we study the ruled surfaces with nonzero constant mean curvature in 3-dimensional Heisenberg group.According to the position relationship between the ruling geodesic and fibers,we divide the problem into several cases.By constructing local parameterizations,the existence of nonzero CMC ruled surface is discussed.For the existing cases,we give the specific parameter expression of the surface and discuss its surface type in R3.We end up with two types of ruled surfaces,one of which is a cylinder,the other of which can be viewed as a surface generated by the screw motion of a cylindrical helix in R3.