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\title{MAT 568:Differential Geometry\\August 30, 2004}
\author{Gabriel C. Drummond-Cole}
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This semester, I'll be trying to condense the material of 568 and 569 into one semester.  That way next semester we can spend on a variety of more advanced topics.  This may require a stronger background on your part.  I'm going to assume you're all familiar with geometry at the level of the core courses, 530 and 531.  I'll use manifolds, bundles, tensors, forms, integrals over them, and so on, without any comment.

The book is Petersen's Riemannian Geometry.  You'll need it, but I won't follow the book very much at all.  You'll be able to see things from many perspectives, which is beneficial for you but also more challenging.  We're going to be looking mainly at the first six chapters.  The later chapters of this book might be the material of 569; we'll have a vote at the end.  If you come across things that seem of interest to you, keep track of them for that purpose.  I've listed some other books on the subject.  

At this level in your experience, there's no such thing as an objectively good book.  All books cover similar material, but from different points of view.  Which books appeal to you will be primarily a matter of taste.

All 500 level classes now have to have nontrivial grading mechanisms, so there will be homework assignments.  There will also be reading assignments.  These will be posted online.
\section{Introduction}
Let me start off with an overview of some of the main issues we're going to cover by the end of the semester.  This subject has a long history, but for the past few decades the main interests have been in curvature and topology.
\begin{itemize}
\item What are the relations between the geometry of a Riemannian manifold $M$ and the topology of $M?$
\item How well can Riemannian metrics "detect" or "see" the topology?
\item Conversely, given a topological manifold $M,$ do Riemannian metrics on $M$ have any special structure related to the topology of $M?$
\item Is there a "best metric" on any manifold?
\item What are geometric invariants of a metric and how do they relate to topology?
\end{itemize}
I'm going to assume you know something about the topology, like I said, and we'll focus on curvature, of which I won't assume knowledge.  There are lots of curvatures of a metric.  There is $Riem,$ the full Riemann curvature tensor, which contains all curvatures.  You want to simplify this somewhat, so you get $Ric,$ the Ricci curvature, which is a function to symmetric bilinear forms.  $Riem$ is a function to a multilinear form with four entries.  $R$ is even simpler, just a number.

All of these are local invariants of a metric.  They only depend on the behavior of a metric near a given point.  They don't depend on the topology, just local properties.  We will do some global things, but these computations are local.

Some global invariants include volume and diameter, the simplest global invariants.
More complicated global invariants involve doing analysis.  You can look at the Laplace operator $\Delta=\sum (\frac{\partial}{\partial x^i})^2,$ the eigenfunctions or eigenvalues of $\Delta,$ geodesics (about which we'll talk a lot), and multi-dimensional analogs, the minimal submanifolds, as well as harmonic maps, and so on.

What are the simplest riemannian manifolds?  The simplest are spaces of constant curvature.  Even though you don't know what curvature is, you're familiar with these spaces.  They're $\mathbb{R}^n$ with the usual Euclidean metric, and quotients by a discrete group of isometries $\mathbb{R}^n/\Gamma,$ for example the $n$-torus $T^n=(S^1)^n.$  So $\mathbb{R}^2/\mathbb{Z}^2=T^2.$  These manifolds have constant curvature 0.

The second class is that of spherical spaces, e.g. $S^n$ with the intrinsic metric from its imbedding in $\mathbb{R}^{n+1},$ and the quotients of these by finite groups. For instance, you can get projective spaces this way.  These manifolds have constant curvature +1.

Then there is constant curvature -1, hyperbolic geometry, $H^n$ and its quotients.

Smartass N. Rounds asks whether $S^1$ is of curvature 0 or 1, and is told that only $M^n$ with $n\ge 2$ has curvature.

\section{"Comparison Geometry"}
We want to compare the geometry of a Riemannian manifold, at least locally, with the geometry of a model space: a flat space, a spherical space, or a hyperbolic space.
\begin{thm}
The Sphere Theorem
\\Suppose that $M^n$ is an $n$-dimensional closed simply connected Riemannian manifold, and at each point $1/4<K\le 1,$ then $M$ is homeomorphic to $S^n.$
\end{thm} 
This result is still being worked on.  It's an open question whether $M$ can have an exotic structure.  There's some higher bound than $1/4$ which implies this, but we don't have optimal bounds.

\section{the "best metric" question}
\begin{itemize}
\item if we can get a constant curvature metric, that's the best.  When $n\ge 3,$ that's too restrictive.
\item this is an area of current research.  People have been working for a long time now on Einstein metrics, where $Ric_g=\lambda\cdot g.$  This ties in with general relativity.
\end{itemize}
The best metric question works best in low dimensions.  When $n=2,$ the uniformization theorem for Riemann surfaces says that every $2$-manifold has a (complete) metric of constant curvature.  This is a classical result of Poincare and Koebe.

For $n=3$ we'd like to do a similar thing.  You can't always get a constant curvature metric.  Thurston's geometrization conjecture says that an analogous statement to uniformization of surfaces holds for all $3$-manifolds.  It looks as though Perelman has proved this, which will prove the Poincare conjecture, which is a purely topological question.  That is, closed simply connected $M^3$ is homeomorphic to $S^3.$

These are the two cases which indicate what we'd like in terms of geometry influencing topology.  In four dimensions there's a lot of work going on; as you get up into higher dimensions things are much less developed.
\section{Riemannian metrics}

Given a smooth $n$-manifold $M^n,\ldots$
\\
I won't harp on this, but here smooth means $C^\infty.$  You could also choose it to be $C^k$ for $k\ge 1.$  There are standard theorems in differential topology that if you have a $C^k$ structure, you have an "equivalent" diffeomorphic $C^\infty$ differential structure as well. 

You have a number of bundles over $M,$ such as $TM.$

$\xymatrix{
\mathbb{R}^n\ar[d]&v\in T_xM\ar[d]
\\ TM\ar[d]^\pi&(x,v)\ar[d]
\\ M&x
}$
\begin{defi}
A Riemannian metric $g$ on $M$ is an assignment or function
\\$x\to g_x,$ a symetric positive definite (i.e., $g_x(v,v)<0,\forall x\ne 0$) bilinear form on $T_xM\equiv \mathbb{R}^n$
\\$M\to S^2(M),$ the space of symmetric bilinear forms on $M.$
which varies continuously or smoothly with $x.$
\end{defi}
What does this mean?  $g$ is a map from $\mathbb{R}^n\times \mathbb{R}^n$ to $\mathbb{R},$ i.e., $(v,w)\to g(v,w)=\langle v,w \rangle.$  As $x$ varies, so do $v=v_x$ and $w=w_x.$

If $X$ and $Y$ are continuous or $C^k$ vector fields then $g(X,Y):M\to\mathbb{R}$ is continuous or $C^k.$

Let $S^2(M)$ be the bundle of symmetric bilinear forms on $M.$
\\$\xymatrix{
S^2_x(M)\ar[d]
\\S^2(M)\ar[d]
\\M}$
\\Here $S^2_x(M)$ is the vector space of symmetric bilinear forms on $V_x=T_xM\equiv \mathbb{R}^n.$
A section of $S^2(M)\to M$ is a function $x\to g_x.$  

Now, just like $TM$ has a standard topology from $\mathbb{R}^n$ and $M,$ so too $S^2(M)$ has a natural topology from the space of symmetric bilinear forms, which is homeomorphic to $\mathbb{R}^{\frac{n(n-1)}{2}},$ and from $M.$  So that's another way to define a Riemannian metric, as a positive definite section of this bundle.

Now, inner products are usually symmetric, so the symmetry in the inner product corresponds to the symmetry here.  We have also the concept of a pseudo-Riemannian metric, where $x\to g_x,$ a symmetric nondegenerate bilinear form.  So we lose positive definiteness.  That is, $det\ g_x\ne 0$ for all $x.$  Any nondegenerate symmetric bilinear form has a basis of $V$ such that $g=\left(\begin{array}{cccccc}+1&&&&&
\\&\ddots&&&&
\\&&1&&&
\\&&&-1&&
\\&&&&\ddots&
\\&&&&&-1
\end{array}\right).$  If there are no $-1$'s then the form is positive definite and you get a Riemannian metric.

The most important pseudo-Riemannian metrics are Lorentzian metrics (general relativity) where you have one negative eigenvalue (time) and any number (often 3) positive eigenvalues (space).

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