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Unformatted text preview: Math 6455 Oct 10, 2006 1 Differential Geometry I Fall 2006, Georgia Tech Lecture Notes 12 Riemannian Metrics 0.1 Definition If M is a smooth manifold then by a Riemannian metric g on M we mean a smooth assignment of an innerproduct to each tangent space of M . This means that, for each p ∈ M , g p : T p M × T p M → R is a symmetric, positive definite, bilinear map, and furthermore the assignment p 7→ g p is smooth, i.e., for any smooth vector fields X and Y on M , p 7→ g p ( X p ,Y p ) is a smooth function. The pair ( M,g ) then will be called a Riemannian manifold. We say that a diffeomorphism f : M → N between a pair of Riemannian manifolds ( M,g ) and ( N,h ) is an isometry provided that g p ( X,Y ) = h f ( p ) ( df p ( X ) ,df p ( Y )) for all p ∈ M and X , Y ∈ T p M . Exercise 0.1.1. Show that the antipodal reflection a : S n → S n , a ( x ) := x is an isometry. 0.2 Examples 0.2.1 The Euclidean Space The simplest example of a Riemannian manifold is R n with its standard Euclidean innerproduct, g ( X,Y ) := h X,Y i . 0.2.2 Submanifolds of a Riemannian manifold A rich source of examples are generated by immersions f : N → M of any manifold N into a Riemannian manifold M (with metric g ); for this induces a metric h on N given by h p ( X,Y ) := g f ( p ) ( df p ( X ) ,df p ( Y )) . In particular any manifold may be equipped with a Riemannian metric since every manifold admits an embedding into R n . 1 Last revised: November 23, 2009 1 0.2.3 Quotient of a Riemannian manifold by a group of isometries Note that the set of isometries f : M → M forms a group. Another source of exam ples of Riemannian manifolds are generated by taking the quotient of a Riemannian manifold ( M,g ) by a subgroup G of its isometries which acts properly discontinu ously on M . Recall that if G acts properly discontinuously, then M/G is indeed a manifold. Then we may define a metric h on M/G by setting h [ p ] := g p . More precisely recall that the projections π : M → M/G , given by π ( p ) := [ p ] is a local diffeomorphism, i.e., for any q ∈ [ p ] there exists an open neighborhood U of p in M and an open neighborhood V of [ p ] in M/G such that π : U → V is a diffeomorphism. Then we may define h [ p ] ( X,Y ) := g q (( dπ q ) 1 ( X ) , ( dπ q ) 1 ( Y )) . One can immediately check that h does not depend on the choice of q ∈ [ p ] and is thus well defined.A specific example of proper discontinuous action of isometries is given by translations f z : R n → R n given by f z ( p ) := p + z where z ∈ Z n . Recall that R n / Z n is the torus T n , which may now be equipped with the metric induced by this group action. Similarly RP n admits a canonical metric, since RP n = S n / {± 1 } , and reflections of a sphere are isometries....
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Tech.
 Spring '08
 Staff
 Geometry

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