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--- math214:02-19 [2020/02/19 01:28]
pzhou created
+++ math214:02-19 [2020/02/19 11:03] (current)
pzhou
@@ Line 13: / Line 13: @@
   - $g_{\alpha \beta} \circ g_{\beta \alpha}=1_F$ over $U_\alpha \cap U_\beta$
   - $g_{\alpha \beta} \circ g_{\beta\gamma} \circ g_{\gamma \alpha} = 1_F$ over $U_\alpha \cap U_\beta \cap U_\gamma$.
+==== Example of Vector Bundles ====
+$E(n,k) \to Gr(n, k)$ tautological bundle over the Grassmannian.
+$O(-1) \to \C\P(1)$.
+==== Operations on Vector Bundles ====
+Direct Sum, Tensor Product, Hom.
 ===== Cotangent Bundle =====
@@ Line 48: / Line 57: @@
 ** Prop **: The cotangent bundle $T^*M = \sqcup_p T_p^*M $ is a vector bundle over $M$.
+===== Poincaré Lemma For 1-form =====
+Given a differential 1-form, we can do line integral.
+A differential 1-form is **closed**, if the value of the line integral is invariant under isotopy that fixes the end-points.
+A differential 1-form $\lambda$ is ** exact**, if there exists a function $f$, such that $df =  \lambda$. We say $f$ is a primitive of $\lambda$.
+Over a unit ball (or more generally, connected space with trivial $\pi_1$), any closed 1-form is exact. We can find a primitive of $\lambda$, by fixing a point $p_0$ in the space, and define $f(p) = \int_{p_0}^p \lambda$.

Lecture Notes