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\centerline{\bf MATHEMATICS 6170}
\centerline{\bf Introduction to Riemannian Geometry}
\centerline{Fall semester 2003}
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\line{ \vbox{
  \hbox{\bf text:\qquad}
  \hbox{}
  \hbox{recommended:\qquad}
  \hbox{}
  \hbox{\bf when:}
  \hbox{\bf where:}}

 \vbox{
  \hbox{\it Riemannian Geometry: A Modern Introduction,}
  \hbox{\qquad by Isaac Chavel, ISBN 0-521-48578-9}
  \hbox{\it Riemannian Geometry}
  \hbox{\qquad by M.P. Do Carmo, ISBN 3-7643-3490-8}
  \hbox{MWF 10:45-11:35}
  \hbox{NS 202}}
 \hfill}
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\line{ \vbox{
  \hbox{\bf instructor:}
  \hbox{\bf \qquad office:}
  \hbox{\bf \qquad telephone:}
  \hbox{\bf \qquad email:}
  \hbox{\bf \qquad office hours:\qquad}
  \hbox{}}
 \vbox{
  \hbox{Prof. Nick Korevaar}
  \hbox{LCB 204}
  \hbox{581-7318}
  \hbox{korevaar{\@}math.utah.edu}
  \hbox{M 2-2:50 p.m.,\ T 10:15-11:15 a.m., 2-2:50 p.m.}
  \hbox{W 2-2:30 p.m.,\ F 8:30-9:30 a.m.,\ or arranged}}
 \hfill}
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\line{{\bf 6170 home page:}\qquad www.math.utah.edu/
$\sim$korevaar/6170fall03\hfill}
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{\bf prerequisites:}  The ideal prerequisite for 
Math 6170 is a course on differentiable
manifolds, like Math 6510 in our catalog.  Alternate prerequisites
for motivated students are the equivalent of both Math 4530 - Euclidean
Curves and Surfaces, and Math 5520 - Introduction to Algebraic and Geometric
Topology. The zeroth chapter of Do Carmo is a 35 page review of
differentiable manifold essentials.
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\noindent
{\bf course outline:} Riemannian Geometry is the 
geometry of curved spaces - precisely 
the study of differentiable manifolds which also have a Riemannian metric - 
an inner product for tangent vectors.  This infinitesimal inner product 
yields infinitesimal elements of arclenth, area and volume and you can 
integrate these to compute  curve lengths, surface areas,  volumes.  
The metric also yields a natural way to differentiate vector fields along
curves, 
called the Riemannian connection.  Riemannian manifolds have a natural
metric space structure:  define the distance between
points to be the length of the shortest curve (geodesic) connecting them.
So, basically, you can do Calculus and analysis on Riemannian Manifolds. 

     The local geometry of Riemannian manifolds is controlled by the
Riemann curvature tensor - for example it affects volume growth of 
balls, whether geodesics bend towards or away from each other, 
the interior angles of geodesic triangles, and it
measures by how much second derivatives fail to commute.  You've
probably seen the effects of this tensor in two dimensions, where it is 
equivalent to Gauss curvature, and where its effects are well-exhibited in 
the constant curvature spaces - the sphere, Euclidean space, and hyperbolic
space.  

     The first part of Math 6170 will expand on the local geometry ideas
sketched in the preceding two paragraphs, chapters 1-4 of Chavel.
  In the second part of the course
we will study global results -  topological, geometric and analytic 
properties of manifolds which arise from pointwise curvature
conditions. These will mostly be drawn from the later chapters of
Chavel and Do Carmo. For example, Preissman's Theorem says that the 
only non-trivial abelian subgroup of the fundamental
group of a negatively curved compact manifold is the integers Z.
Another example is the sphere theorem:  If the Riemannian sectional
curvature of a compact manifold is bounded strictly between
1/4 and 1, then the manifold is topologically an n-sphere.  This theorem is
sharp.

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{\bf grading:}\qquad Homework will be assigned regularly.
You will provide written solutions, and we may have a weekly
problem session as well at which these can be discussed.
You will also be responsible for a final project -  the presentation to the
class of an interesting paper or result from Riemannian geometry.
If you already know your area of specialization we may be able to
find a paper in that area which uses a Riemannian setting.  Otherwise, 
 there are a number of global theorems which are accessible.
