# Tagged Questions

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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### complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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### Ring of smooth functions on a manifold and localization with respect to a multiplicative system

Take $X$ a smooth manifold and $x\in X$. It can be shown that the germ of smooth functions around $x$, $C^\infty(X)_x$ is equal to the algebraic $S^{-1}C^\infty (X)$ where $S$ is the set of smooth ...
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### Classification of Möbius Transformations

We know how to classify the points on a surface,by looking the Gaussian curvature at a point in order to guess the shape of the surface near that point.On the other hand we classify the Möbius ...
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### The Lie Algebra of $O(n)$ is the set of $n \times n$ skew-symmetric matrices

I'm trying to show that the Lie Algebra for $O(n)$ is the set of $n \times n$ skew-symmetric matrices. Here is what I have so far. Since $O(n)$ is the union of two disjoint subsets, the matrices with ...
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