Contents

22 2 Differential geometry without a metrica map Φ ∗ t T of any tensor T , called Lie transport. The Lie derivative ofT with respect to v is defined by1LvT ≡ limt→0 t (Φ∗ t T − T ). (2.56)The tensors T and Φ ∗ t T are of the same type (r, s) and are both evaluatedat the same point p. Therefore, the Lie derivative (2.56) is also a tensorof type (r, s) atp. The Lie derivative vanishes if the tensors T and Φ ∗ t Tcoincide. In this case the tensor field T remains in a sense the ‘same’ underLie transport along the integral curves of the vector field v. However,the components of T with respect to the coordinate basis {∂/∂x i } mayvary along the curves. Using coordinate bases {∂/∂x i } and {∂/∂y i },wecompute the components of the Lie derivative. The relations∂y i∂x k ∣ ∣∣∣∣t=0= δ i k,dy idt= v i ,∣ t=0dx idt= −v i (2.57)∣ t=0will be used. We start with the Lie derivatives of functions, 1-forms, andvectors:function f: Lvf = v i f, i (= v(f)). (2.58)Proof:Φ ∗ t f| p = f(y(x, t)),Lvf| p = ∂fdy i∂y i .dt ∣ p1-form σ: Lvσ =(v m σ i,m + σ m v m ,i)dx i . (2.59)Proof:vector u:Proof:Φ ∗ t σ| p = σ j (y(x, t)) ∂yj∂x i dxi ,[∂σj dy m ∂y j ( )]Lvσ| p =∂y m dt ∂x i + σ ∂ dyjj∂x i dtt=0dx i .Lvu =(v m u i ,m − u m v i ,m) ∂∂x i . (2.60)Φ ∗ t u| p = u j (y(x, t)) ∂xi ∂∂y j ∂x i ,[∂ujdy m ∂x i ( )]Lvu| p =∂y m dt ∂y j + ∂ dxiuj∂y j dtt=0∂∂x i .