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\begin{document}


HW  Assignment 1.  

1.1 A. Prove that  the locus $xy =0$ in the plane, is NOT a topological manifold.

 B. For what values of $c$ is the locus $xy =c$ a smooth manifold? 

\vskip .3cm

  \vskip .2cm 
  
  1.2. Set-up: Consider the space of all lines in the plane. 
 Since a line can be given in slope-intercept form: 
 $y = mx + b$, we can think of the slope m and intercept
 b as the coordinates of the line.  Thus the space of lines must form a
  two-dimensional manifold.   But the slope-intercept form misses one family of lines, the vertical lines,
  which have infinite slope.  We can write these lines in the form
  $x = B$, B a constant, and as such they  form part of the family of lines expressed in the form
  $x = My + B$.
  
 1.2 A.  Compute the overlap map $(m,b) \mapsto (M, B)$.
  
  1.2B.  Prove that the space of all lines in the plane , with the manifold structure you just deduced,
  is diffeomorphic to the tangent bundle of the circle, and that this tangent bundle is
  in turn diffeomorphic to the cylinder $S^1 \times \R$.
  
  \vskip .3cm 
  
1.3.  A holomorphic function $F: \C \to \C$ can be viewed as a function
from the plane $\R^2$ to itself, by having  $z = x + iy \in \C$ correpsond to $(x,y) \in \R^2$.
What is the relation between the complex derivative $dF/dz (z) \in \C$ and the real 
linearization, which is a linear map $\R^2 \to \R^2$?  

\vskip .3cm
1.4. Let $gl(n)$ denote the vector space of n by n real matrices, 
and $Sym(n)$ the vector space of symmetrid n by n real matrices.

a) What is the dimension of $gl(n)$?

 

b) What is the dimension of $Sym(n)$?

c) Compute the derivative of the map  $F(A) = A A ^t$ at the value $A = Id$.  

d) Think of this $F$ as a map  $F: gl(n) \to Sym(n)$
 Show that $Id \in Sym(n)$ is a regular VALUE of $F$.

e)  Conclude that  $F^{-1} (Id)$ is a smooth manifold
and compute its dimension.  (Terminology:  This level set $F^{-1} (Id)$ is known as 
the  orthogonal group and denoted by $O(n)$. It is the group of linear isometries $\R^n \to \R^n$ .)
 

\end{document}
