### Albanese Map of a Riemannian Manifold

Differential Geometry

Algebraic Geometry

Given a compact Riemannian manifold \(M\) such that the first Betti number \(b\) is nonzero, there exists a map \(\operatorname{alb}: M\to \mathbb{R}^b/\mathbb{Z}^b\), called the Albanese map. The aim of post is to prove the smoothness of the Albanese map.

### A Brief Introduction to Cohomology of Lie Algebra

Differential Geometry

Let \(G\) be a compact simply connected Lie group and \(\mathfrak{g}\) its Lie algebra. In this posts, we will discuss the motivation of Lie algebra cohomology of \(\mathfrak{g}\) and its connection with the de Rham cohomology of \(G\).

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