Math 240A: Differentiable Manifolds and
               Riemannian Geometry
                Simon Rubinstein–Salzedo
                    Fall 2005
            0.1  Introduction
            These notes are based on a graduate course on differentiable manifolds and Rieman-
            nian geometry I took from Professor Doug Moore in the Fall of 2005. The text-
            books were An Introduction to Differentiable Manifolds and Riemannian Geometry
            by William Boothby and Calculus on Manifolds by Michael Spivak. Many other
            books are also mentioned in the notes. Since the professor handed out very good
            notes, I have made very few changes to these notes.
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               Chapter 1
               September 22, 2005
               Calculus of several variables starts with calculus on Rn and then proceeds to “calculus
               on manifolds.”
               Let U be an open subset of Rn and f : U → R a function. f is C0 iff f is continuous.
               f is Ck iff f has continuous partial derivatives of order at most k. f is C∞ iff f
               has continuous partial derivatives of all orders. f is Cω iff f can be extended in a
               convergent Taylor series expansion about any point.
               Suppose F : U → Rn, F = (f1,...,fn). F is C0, Ck, C∞, or Cω iff each fi is.
               Let M be a topological space. An n-dimensional chart or coordinate system on
               M is a pair (U,ϕ) such that U is an open subset of M and ϕ is a homeomorphism
               from U onto an open subset ϕ(U) of Rn.
               Let r ∈ {0,1,2,...,k,...,∞,ω}. An n-dimensional Cr-atlas on M is a collection
               A={(Uα,ϕα):α∈A}ofn-dimensional charts such that
                 1. M =S{Uα :α∈A}.
                 2. ϕβ ◦ ϕ−1 is a Cr map where defined for all α,β ∈ A.
                         α
               Two n-dimensional Cr atlases A and A are equivalent if A ∪A is a Cr-atlas.
                                          1     2                 1   2
                                                 2
                            Definition. An n-dimensional Cr-manifold is a Hausdorff second countable topo-
                            logical space M together with an equivalence class of Cr-atlases.
                            Examples.
                                 1. Let U be an open subset of Rn. Then {(U,id)} is a Cr-atlas making U into an
                                     n-dimensional Cr-manifold, for any r.
                                               n             1          n+1             n+1           1 2                    n+1 2
                                 2. Let S          = {(x ,...,x               ) ∈ R            :  (x ) + ··· + (x                 )    = 1}. Let n =
                                     (0,...,0,1), s = (0,...,0,−1), U = Sn − {n}, and V = Sn − {s}. Define
                                     ϕ:U →Rn by
                                                         ϕ(x1,...,xn+1) =                   1       (x1,...,xn) = (u1,...,un).
                                                                                      1−xn+1
                                     Then (u1,...,un,0) = (0,...,0,1) + λ(x1,...,xn+1 − 1), where λ =                                                  1    . ϕ
                                                                                                                                                        n+1
                                                                                                                                                   1−x
                                     is a homeomorphism from U to Rn because we can solve for (x1,...,xn+1) in
                                     terms of (u1,...,un):
                                                                                    1 2                  n 2                 n+1 2                 n+1
                                                 1 2                  n 2       (x ) +···+(x )                     1−(x           )       1+x
                                              (u ) +···+(u ) =                                  n+1 2          =             n+1 2 =               n+1.
                                                                                      (1 −x          )             (1 −x          )       1−x
                                                   2         n+1               n+1           n+1            2            2                    n+1        |u|2−1
                                     Thus|u| (1−x                 ) = 1+x           , so x        (1+|u| ) = |u| −1. Hence x                        =|u|2+1,
                                     and 1−xn+1 =                 2    ,
                                                               |u|2+1
                                                  (x1,...,xn) = (1−xn+1)(u1,...,un) =                                 2       (u1,...,un)
                                                                                                                 |u|2 + 1
                                                   1          n+1           −1     1          n             1            1            n       2
                                               (x ,...,x           ) = ϕ       (u ,...,u ) =               2       (2u ,...,2u ,|u| −1).
                                                                                                       |u| +1
                                     Similarly, we define ψ : V → Rn by
                                                       ψ(x1,...,xn+1) =                   1       (x1,...,xn+1) = (v1,...,vn).
                                                                                     1+xn+1
                                     WeclaimthatA={(U,ϕ),(V,ψ)}isaCratlasmakingSnintoann-dimensional
                                     Cr-manifold. Indeed,
                                                                                      1−xn+1                                 1
                                                   ψ◦ϕ−1(u1,...,un) =                         n+1(u1,...,un) =                 2(u1,...,un),
                                                                                      1+x                                  |u|
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