18.966 Geometry of Manifolds: Lecture 1

Department of Mathematics 18.965 Fall 04 Lecture Notes Tomasz S. Mrowka Lecture 1. 1 Manifolds: definitions and examples Loosely manifolds are topological spaces that look locally like Euclidean space. A little more precisely it is a space together with a way of identifying it locally with a Euclidean space which is compatible on overlaps. To formalize this we need the following notions. Let X be a Hausdorff, second countable, topological space. Definition 1.1. A chart is a pair (U, φ) where U is an open set in X and φ : U → Rn is homeomorphism onto it image. The components of φ = (x 1 , x 2 , . . . , x n ) are called coordinates. Given two charts (U1 , φ1 ) and (U2 , φ2 ) then we get overlap or transition maps φ2 ◦ φ1−1 : φ1 (U1 ∩ U2 ) → φ2 (U1 ∩ U2 ) and φ1 ◦ φ2−1 : φ2 (U1 ∩ U2 ) → φ1 (U1 ∩ U2 ) Definition 1.2. Two charts (U1 , φ1 ) and (U2 , φ2 ) are called compatible if the over­ lap maps are smooth. In practice it is useful to consider manifolds with other kinds of regularity. One many consider C k ­manifolds where the overlaps are C k ­maps with C k inverses. If we only require the overlap maps to be homeomorphisms we arrive at the notion of a topological manifold. In some very important work of Sullivan one consider Lipschitz, or Quasi­conformal manifolds. 1 An atlas for X is a (non­redundant) collection A = {(Uα , φα )|α ∈ A} of pair wise compatible charts . Two atlases are equivalent if there their union is an atlas. An atlas A is called maximal if any other atlas compatible with it is contained in it. Exercise 1. Using Zorn’s lemma, show that any atlas is contained in a unique maximal atlas. 2 Definition 1.3. A smooth n­dimensional manifold is a Hausdorff, second count­ able, topological space X together with an atlas, A. 1.1 examples Rn or any open subset of Rn is a smooth manifold with an atlas consisting of one chart. The unit sphere S = {(x , x , . . . , x )| n 0 1 n n � (x i )2 = 1} i=0 has an atlas consisting of two charts (U± , φ± ) where U± = S n \{(±1, 0, 0, . . . , 0)} and φ± (x 0 , x 1 , . . . , x n ) = 1 (x 1 , . . . , x n ) ±1 − x0 Real projective space, RPn , is space of all lines through the origin in Rn+1 which we can identify with nonzero vectors up to the action of non­zero scalars so RPn = (Rn+1 \ {0�})/R∗ . The equivalence class of (x0 , . . . , xn ) is denoted [x0 : x1 : . . . : xn ]. RPn has an atlas consisting of n + 1 charts. The open sets are Ui = {[x0 : x1 : . . . : xn ]|x j ∈ R, and xi � = 0} and the corresponding coordinates are φi ([x0 : x1 : . . . : xn ]) = (x1 /xi , . . . , x� i /x i , . . . , x n /x i ). Similarly we have complex projective space, CPn , the space of a line through the origin in Cn+1 . So just as above we have CPn = (Cn+1 \ {0�})/C∗ . A typical point of CPn is written [z 0 : z 1 : . . . : z n ]. CPn has a atlas consisting of n + 1 charts. The open sets are Ui = {[z 0 : z 1 : . . . : z n ]|z i � = 0} and the corresponding coordinates are φi ([z 0 : z 1 : . . . : z n ]) = (z 1 /z i , . . . , z� i /z i , z n /z i ). . 3 Exercise 2. Show that in fact the above construction yield charts. Notice that in the case of CPn the coordinates have values in Cn and so the overlap maps map an open subset of Cn to Cn . We can ask that they are holomorphic. We make the following definition. Definition 1.4. A complex manifold is a Hausdorff second countable topological space X , with an atlas A = {(Uα , φα )|α ∈ A the coordinate functions φα take values in Cn and so all the overlap maps are holomorphic. Let Grk (Rn ) be the space of k­planes through the origin in Rn . � � Exercise 3. Show that Grk (Rn ) has an atlas with nk charts each homeomorphic with Rk(n−k) . Similarly we have Grk (Cn ) the space of all complex k­plane through the origin in Cn . � � Exercise 4. Show that Grk (Cn ) has an atlas with nk charts each homeomorphic with Ck(n−k) . Show that we can give Grk (Cn ) the structure of a complex manifold. 4