By Gudmundsson S.

Those lecture notes grew out of an M.Sc. path on differential geometry which I gave on the collage of Leeds 1992. Their major function is to introduce the gorgeous idea of Riemannian Geometry a nonetheless very energetic study quarter of arithmetic. it is a topic without loss of attention-grabbing examples. they're certainly the main to an exceptional figuring out of it and may for this reason play a tremendous function all through this paintings. Of particular curiosity are the classical Lie teams permitting concrete calculations of the various summary notions at the menu.

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Extra info for An Introduction to Riemannian Geometry

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62 6. 1. Let M be a smooth manifold and ∇ nection on the tangent bundle (T M, M, π). Prove that the torsion ˆ is a tensor field of type (2, 1). 2. 8. 3. 9. 4. Let SO(m) be the special orthogonal group equipped with the metric 1 X, Y = trace(X t · Y ). 2 Prove that , is left-invariant and that for left-invariant vector fields X, Y ∈ so(m) we have ∇XY = 12 [X, Y ]. Let A, B, C be elements of the Lie algebra so(3) with       0 −1 0 0 0 −1 0 0 0 Ae =  1 0 0  , Be =  0 0 0  , Ce =  0 0 −1  .

Dt dt Then |dφq (γ˙ w (0))| = |qwi| = |q||w| = 1 implies that the differential dφq is injective. It is easily checked that the immersion φq is an embedding. dφq (γ˙ w (0)) = In the next example we construct an interesting embedding of the real projective space RP m into the vector space Sym(Rm+1 ) of the real symmetric (m + 1) × (m + 1) matrices. 21. Let m be a positive integer and S m be the mdimensional unit sphere in Rm+1 . For a point p ∈ S m let Lp = {(s · p) ∈ Rm+1 | s ∈ R} be the line through the origin generated by p and ρp : Rm+1 → Rm+1 be the reflection about the line Lp .

10. Let M be a manifold and X, Y ∈ C ∞ (T M ) be vector fields on M . Then the section [X, Y ] : M → T M of the tangent bundle given by [X, Y ] : p → [X, Y ]p is smooth. Proof. 9 that the section [X, Y ] is smooth. For later use we prove the following useful result. 11. Let M be a smooth manifold and [, ] be the Lie bracket on the tangent bundle T M . Then (i) [X, f · Y ] = X(f ) · Y + f · [X, Y ], (ii) [f · X, Y ] = f · [X, Y ] − Y (f ) · X for all X, Y ∈ C ∞ (T M ) and f ∈ C ∞ (M ), Proof. If g ∈ C ∞ (M ), then [X, f · Y ](g) = X(f · Y (g)) − f · Y (X(g)) = X(f ) · Y (g) + f · X(Y (g)) − f · Y (X(g)) = (X(f ) · Y + f · [X, Y ])(g) This proves the first statement and the second follows from the skewsymmetry of the Lie bracket.