By Gudmundsson S.
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Additional resources for An introduction to Riemannian geometry
Dφ)p (γ(0)) ˙ = This means that and dφp (a) = (q → 2 q, a p + 2 q, p a) dφp (b) = (q → 2 q, b p + 2 q, p b). If a = b then dφp (a) = dφp (b) so the differential dφp is injective, hence the map φ : S m → R(m+1)×(m+1) is an immersion. If the points p, q ∈ S m are linearly independent, then the lines Lp and Lq are different. But these are just the eigenspaces of ρp and ρq with the eigenvalue +1, respectively. This shows that the linear endomorphisms ρp , ρq of Rm+1 are different in this case. On the other hand, if p and q are parallel then p = ±q hence ρp = ρq .
4. THE TANGENT BUNDLE 39 Proof. 4. If φ : M → N is a surjective map between differentiable manifolds ¯ ∈ C ∞ (T N ) are said to be then two vector fields X ∈ C ∞ (T M ), X ¯ φ(p) for all p ∈ M . In that case we write φ-related if dφp (X) = X ¯ X = dφ(X). 14. Let φ : M → N be a map between differentiable ¯ Y¯ ∈ C ∞ (T N ) such that X ¯ = dφ(X) manifolds, X, Y ∈ C ∞ (T M ), X, and Y¯ = dφ(Y ). Then ¯ Y¯ ] = dφ([X, Y ]). [X, Proof. Let f : N → R be a smooth function, then ¯ Y¯ ](f ) = dφ(X)(dφ(Y )(f )) − dφ(Y )(dφ(X)(f )) [X, = X(dφ(Y )(f ) ◦ φ) − Y (dφ(X)(f ) ◦ φ) = X(Y (f ◦ φ)) − Y (X(f ◦ φ)) = [X, Y ](f ◦ φ) = dφ([X, Y ])(f ).
A vector field X ∈ C ∞ (T G) is said to be left invariant if dLp (X) = X for all p ∈ G, or equivalently, Xpq = (dLp )q (Xq ) for all p, q ∈ G. The set of all left invariant vector fields on G is called the Lie algebra of G and is denoted by g. The Lie algebras of the classical Lie groups introduced earlier are denoted by gl(Rm ), sl(Rm ), o(m), so(m), gl(Cm ), sl(Cm ), u(m) and su(m), respectively. 18. Let G be a Lie group and g be the Lie algebra of G. e. if X, Y ∈ g then [X, Y ] ∈ g, Proof. If p ∈ G then dLp ([X, Y ]) = [dLp (X), dLp (Y )] = [X, Y ] for all X, Y ∈ g.
An introduction to Riemannian geometry by Gudmundsson S.