Some notation to be used:
-n dimensional Euclidean space, $Bbb R^n$
-n dimensional complex space, $\Bbb C^n$
-The general linear group, $\text{GL}(n, \Bbb R)$.
-In general, any classical group should be partially in text mode. That
is, the unitary group would be $\text{U}(n)$, the symplectic group
would be $\text{Sp}(n)$, etc.
-The preimage of a trivial nbhd, $E|U_\alpha$
-The target of a trivialization, $U_\alpha \times \Bbb R^n$
-A transition function, $g_{\alpha \beta}$
-The identity map $\text{Id}$ (in any setting, including in the general linear
group).
-Examples should be set using the \example,\endexample construct in amstex.
Examples should be numbered.
-Similarly for defintions, use the \definition, \enddefinition construct.
Definitions should be numbered.
-For exercises, use the \remark,\endremark construct. These should be
numbered too.
-A set of stuff: $\{U_\alpha \}$
-Formal proofs should use the \proclaim construct.
-When you start a formal proof, use {\bf Proof:}
-When a formal proof is finished, use \qed at the end to get the little box
marking the end of the proof.
Example:
\proclaim{A Silly Theorem} The complex exponential function constant.
\endproclaim
{\bf Proof:} On $\Bbb C-0$, the maximum modulus theorem asserts
that $e^z$ attains a maximum modulus on the boundary. That is,
$e^0=1$ is the maximum modulus of the exponential funciton. By
Liouville's Theorem, $e^z$ is constant.\qed
-The action of a group on a space $x \cdot g$.
-The composition of functions:Either $fg$, or where necessary for
understanding: $\f \circ g$.
-Categories should be in bold if named, script if lettered. That is,
{\bf Smooth} is the category of smooth manifolds. $\Cal{C}$ is
a general category. The distinguished categories we have are
{\bf Smooth}, {\bf TopM} - topological manifolds, {\bf Holomorphic} -
holomorphic manifolds, {\bf Top} - topological spaces, {\bf Ab} -
- Abelian groups.
-Isomorphism is denoted by $\cong$. I.e, $A \cong B$.
-Homotopy is denoted by $\simeq$. I.e, $f \simeq g$.
-Relations are denoted by $\sim$. I.e, $x \sim y$.
-Homeomorphism/Diffeomorphism is denoted by $\approx$. I.e, $X \approx Y$.
-Use blackboard bold for the notation of the standard set/rings/fields, $\Bbb N, \Bbb Z, \Bbb R, \Bbb Q, \Bbb C,\Bbb H$.
-The various projective spaces are $\Bbb RP^n$, $\Bbb CP^n$, and $\Bbb HP^n$.