The Classification Of Quadratic Surfaces In Three-Dimensional Minkowski Space

Two hundred years ago, human knew little about Non-Euclidean space. In that time, Euclidean space, which can reflect the real world to some extents, has been thought the only right geometry space. At the beginning of the nineteenth century mathematicians such as C.F.Guass, Janos Bolyai and Lobatchvski found the reasonability of the pseudo-Euclidean space. After that, pseudo-Euclidean space has become an important topic. But the three-dimensional Minkowski space is most widely researched because it has only one negative index and a good symmetry that is more near to the three-dimensional Euclidean space than other pseudo-Euclidean spaces.In three-dimensional Euclidean space, it is well known that there are seventeen different quadratic surfaces that are classified by invariance of surfaces under the rotations and parallel translations of coordinate axis. Similarly, do Carmo defined the rotations of coordinate axis in Lorentz space, and we can define the same parallel translations of coordinate axis in Lorentz space as that defined in Euclidean space. So in Lorentz space we can also classify the quadratic surfaces depending on the rotations and parallel translations of coordinate axis.At first, in this thesis, we consider the invariants of quadratic surfaces under the rotations and parallel translations of coordinate axis in three-dimensional Minkowski space. Then we can define the equivalent relation of quadratic surfaces according to these invariants. Finally we can classify the quadratic surfaces with the equivalent relation in three-dimensional Minkowski space.