On Calculus of Manifolds with Special Emphasis of 3D Minkowski Space M2,1

Anis I Saad

In this paper, we explain some topics of calculus of manifold, especially for the spacetime symmetry topic. With emphasis of 3D Minkowski differential geometry. The most important symmetries are  , A diffeomorphism of this symmetry is called the isometry. If a one-parameter group of isometries is generated by a vector field, then "this vector field is called a Killing vector field. Which shows that the Lei derivative is vanishing" [14]. Moreover the one parameter group of diffeomorphism called the flow. However the Poincare' group "is the group of isometries of Minkowski spacetime. Also "it is a full symmetry of special relativity includes the translations , rotation and boosts" [11].