Download A Comprehensive Introduction to Differential Geometry, Vol. by Michael Spivak PDF

By Michael Spivak

Publication via Michael Spivak, Spivak, Michael

Show description

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

Best differential geometry books

Development of the Minkowski Geometry of Numbers Volume 2

Sooner than those very good expositions, Minkowski's pioneering writings have been available simply to experts. This vintage two-volume paintings focuses totally on geometric difficulties related to integers and algebraic difficulties approachable via geometrical insights. It demonstrates the simplicity and magnificence of quantity idea proofs and theorems and illuminates many different algebraic and geometric issues.

Handbook of Geometric Analysis,

Geometric research combines differential equations with differential geometry. an enormous point of geometric research is to process geometric difficulties by way of learning differential equations. in addition to a few recognized linear differential operators resembling the Laplace operator, many differential equations bobbing up from differential geometry are nonlinear.

The Riemann Legacy: Riemannian Ideas in Mathematics and Physics

The learn of the increase and fall of significant mathematical principles is unquestionably the most attention-grabbing branches of the background of technological know-how. It allows one to return into touch with and to take part on the planet of rules. Nowhere will we see extra concretely the big non secular power which, at first nonetheless missing transparent contours, begs to be moulded and built by means of mathematicians, than in Riemann (1826-1866).

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.

Download PDF sample

Rated 4.77 of 5 – based on 49 votes