By Michael Spivak

Publication via Michael Spivak, Spivak, Michael

**Read or Download A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition PDF**

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**Additional resources for A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition**

**Example text**

3. Let X = G/K be a symmetric space of noncompact type. If τ : G → S L(n, R) is a faithful representation satisfying τ (θ (g)) = (τ (g)t )−1 , where θ is the Cartan involution of G associated with K , then the map i τ : X → Pn , gK → gg t embeds X as a totally geodesic submanifold of Pn . 26 L. Ji Briefly, the condition τ (θ (g)) = (τ (g)t )−1 implies that K is mapped into S O(n). Then the map gK → gg t is well-defined. Since the representation τ is faithful, this map is an embedding. The above condition again implies that the image i τ (X ) is a totally geodesic submanifold of Pn .

2. If g is semisimple, n(k) = k. 34. Let (g, σ ) be an irreducible, reduced, orthogonal involutive Lie algebra, Q a positive definite quadratic form on p invariant under k, c the constant such that B|p = cQ. Then there are three possibilities: 1. c = 0, and (g, σ ) is flat and g has no semisimple ideal. 2. c > 0, g is simple and noncompact and k is a maximal compact subalgebra. 3. c < 0, g is compact and either g is simple or g = g1 × g1 , where g1 is simple and σ (X, Y ) = (Y, X ), where X, Y ∈ g1 .

For any X ∈ a⊥ , Y ∈ g, Z ∈ a, < [X, Y ], Z > = Re T r (X Y Z ∗ − Y X Z ∗ ) = −Re T r (X Z ∗ Y − X Y Z ∗ ) = −Re T r (X (Y ∗ Z )∗ − X (Z Y ∗ )∗ ) = −Re T r (X (Y ∗ Z − Z Y ∗ )∗ ) = − < X, [Y ∗ , Z ] >= 0. In the last equation, we used the fact that a is an ideal and Y ∗ ∈ g. Since Z ∈ a is arbitrary, [X, Y ] ∈ a⊥ . In the above proposition, the converse is also true with respect to a suitable basis. An immediate corollary is that many classical Lie algebras are reductive. For example, u(n) = {X ∈ gl(n, C) | X + X ∗ = 0} is clearly closed under X → X ∗ and hence reductive.