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This is probably not "down-to-Earth" enough for your purposes, but it was one of the first uses of de Rham cohomology that I really enjoyed and I feel like I must share it. (I learned it from Bott's a …

The question has already been answered by Bugs Bunny, but I thought I'd point out that there is a nice paper by H.-C. Wang from the 1950s that discusses the complex structure of homogeneous manifolds …

Is there an example of a smooth $n$-manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$? (This ques …

You can handle the case of $n \leq 3$ one at a time, and so the question really is about $n \geq 5$. Two important names in this regard are Kirby and Siebenmann. The Wikipedia article on the Hauptverm …

Using the maximum modulus principle you can show that $\mathbb{C}^n$ doesn't have any compact complex submanifolds of positive dimension. It follows that lots of complex manifolds, such as complex gra …

I think there's some confusion in the question. For example by "Levi-Civita connection" you must really mean some kind of Laplacian. Anyway, your end result about the cohomology of $G/H$ is essentiall …

Although the title is about Lie algebras, the question body mentions Lie groups, and my answer will deal more with these. As mentioned in other answers, Lie groups show up frequently in geometry as gr …