1
Lecture 1 (Notes: K. Venkatram)
1.1
Smooth Manifolds
Let M be a f.d. C manifold, and C (M ) the algebra of smooth R-valued functions. Let T = T M be
the tangent bundle of M : then C (T ) is the set of derivations Der(C (M ), i.e. the set of morphisms
X
2
2.1
Lecture 2 (Notes: A. Rita)
Comments on previous lecture
(0) The Poincar lemma implies that the sequence
e
d
d
. . . C (k1 T ) C (k T ) C (k+1 T ) . . .
is an exact sequence of sheaves, even though it is not an exact sequence of vector spaces.
(1) We
4
4.1
Lecture 4 (Notes: J. Pascale )
Geometry of V V
Let V be an n-dimensional real vector space, and consider the direct sum V V . This space has a natural
symmetric bilinear form, given by
X + , Y + =
1
( (Y ) + (X )
2
for X, Y V , , V . Note that the
5
Lecture 5 (Notes: C. Kottke)
5.1
Spinors
We have a natural action of V V on
V . Indeed, if X + V V and
V , let
(X + ) = iX + .
Then
(X + )2
= iX (iX + ) + (iX + )
= (iX ) iX + iX
= X + , X +
where , is the natural symmetric bilinear form on V V :
1
9
Lecture 9 (Notes: K. Venkatram)
Last time, we talked about the geometry of a connected lie group G. Specically, for any a in the
corresponding Lie algebra g, one can dene aL |g = Lg a and choose L 1 (G, g) s.t. L (aL ) = a. For
instance, for GLn , with