By Karl-Heinz Fieseler
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Additional info for Riemannian Geometry [Lecture notes]
Sample text
Let f : Q −→ M be a differentiable map. A section of E above or over f is a differentiable map µ : Q −→ E with µq ∈ Ef (q) for all q ∈ Q. E µ ↓ f Q −→ M 2. We denote Γf (E) the vector space of all sections of E above f , a C ∞ (Q)module. 3. For E = T M we write Θf (M ) := Γf (T M ). 5. If D is a connection on E we may define DX µ, where X ∈ Θ(Q) is a vector field on Q and µ a section of E above f . , µn ∈ Γ(E|U ) in an open neighbourhood U of f (Q). 4. This definition is independent from the chosen frame and thus can be used locally in order to patch together the definitions on the members of an open cover of f (Q) in the general case.
N That implies σM ∼ =σ 8 n P2 (C) ∼ = In . Connections on T M Denote ∇ : Θ(M ) × Θ(M ) −→ Θ(M ) a connection on the tangent bundle of the differentiable manifold M . 1. The torsion tensor field T∇ ∈ Γ(T 1,2 M ) of the connection ∇ is defined by T∇ (X, Y ) := ∇X Y − ∇Y X − [X, Y ]. We call ∇ torsion free if T∇ = 0. 2. , xm , we have (Γkij − Γkji ) ∂k ⊗ dxi ⊗ dxj . , m. 3. Let M be a pseudo-Riemannian manifold. Then there is a unique torsion free ”metric” connection ∇ on T M , called the Levi-Civitaconnection.
Obviously all paths t → f (t, w) are geodesics with tangent vectors of length ||Xa ||. We want to apply the below lemma with T := T f (∂t ), W = T f (∂w ) ∈ Θf (M ). So we have to compute g(W, T ). We obtain W(t,w) = T expa (t(− sin(w)Xa + cos(w)Ya )). In particular W(t,0) = T expa (tYa )) and thus, according to Rem. 13, g(W(t,0) , γ(t)) ˙ = g(W(0,0) , γ(0)) ˙ = g(0, γ(0)) ˙ = 0. With t = 1 we obtain TXa expa (Ya ) ⊥ γ(1). 13. e. all curves t → f (t, w) are geodesics. If in addition they have tangent vectors of the same length, then for T := T f (∂t ), W := T f (∂w ) ∈ Θf (M ) the inner product g(W, T ) : I × (−ε, ε) −→ R does not depend on t.