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 attractive concept of Riemannian Geometry a nonetheless very lively examine sector of arithmetic. this can be a topic with out loss of attention-grabbing examples. they're certainly the foremost to a very good figuring out of it and should for this reason play an enormous function all through this paintings. Of precise curiosity are the classical Lie teams permitting concrete calculations of a number of the summary notions at the menu.

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This means that the 3-dimensional sphere S 3 is disjoint union of great circles S3 = φ−1 ({q}). q∈S 2 32 3. 1. Let p be an arbitrary point on the unit sphere S 2n+1 of Cn+1 ∼ = R2n+2 . Determine the tangent space Tp S 2n+1 and show that it contains an n-dimensional complex subspace of Cn+1 . 2. 3. Prove that the matrices 1 0 0 −1 X1 = , X2 = , 0 −1 1 0 X3 = 0 1 1 0 form a basis for the tangent space Te SL(R2 ) of the real special linear group SL(R2 ) at e. For each k = 1, 2, 3 find an explicit formula for the curve γk : R → SL(R2 ) given by γk : s → Exp(s · Xk ).

Let (M, g) be a Riemannian manifold. Then the Levi-Civita connection ∇ is a unique metric and torsion free connection on the tangent bundle (T M, M, π). Proof. 4. This is the same as 1 = {g(Z, [X, Y ]) − g(Z, [Y, X])} = g(Z, [X, Y ]). 2 This proves that the Levi-Civita connection is torsion-free. 4. This is exactly 1 = {X(g(Y, Z)) + X(g(Z, Y ))} = X(g(Y, Z)). 2 This shows that the Levi-Civita connection is compatible with the Riemannian metric g on M . 6. 7. Let G be a Lie group. For a left invariant vector field Z ∈ g we define the map ad(Z) : g → g by ad(Z) : X → [Z, X].

It is not difficult to see that ˆ B = {(π −1 (U ), x¯)| (U, x) ∈ A} 36 4. THE TANGENT BUNDLE is a bundle atlas making (T M, M, π) into an m-dimensional topological vector bundle. It immediately follows from above that (T M, M, π) together with the maximal bundle atlas Bˆ defined by B is a differentiable vector bundle. The set of smooth vector fields X : M → T M is denoted by C ∞ (T M ). 6. We have seen earlier that the 3-sphere S 3 in H ∼ = C2 carries a group structure · given by ¯ zβ + wα (z, w) · (α, β) = (zα − wβ, ¯ ).