# Questions tagged [semi-riemannian-geometry]

It is the study of smooth manifolds equipped with a non-degenerate metric tensor, not necessarily positive-definite (and hence a generalisation of [riemannian-geometry]). Included in this are metric tensors with index 1, called "Lorentzian", which are used to model spacetimes in (general-relativity).

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### Local cartesian coordinates on Riemannian manifold

I'm wondering is possible for every given metric $g=g_{ij}dx^i \otimes dx^j$ on $M$ and for every given $p\in M$ to find such chart $(U, \varphi)$ around $p\in U \subset M$ that the metric $g|_U$ in ...
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### Boundary points of a smooth manifold with boundary independent of chart.

Setup: Let $$\mathbb{R}^n\_ := \{(x^1,\ldots,x^n) \subset \mathbb{R}^n: x^1 \leq 0\}$$ and $$\partial \mathbb{R}^n\_ := \{0\} \times \mathbb{R}^{n-1}$$ i.e. $$x \in \partial \mathbb{R}^n\_$$ are on ...
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### Integral of an $n$-form with compact support.

Setup: Let $M^n$ be an oriented manifold and let $$\mathcal{A} := \{(U_i,\varphi_i):i \in I\}$$ be a positively oriented atlas ($\varphi_i:U_i \to \varphi_i(U_i)$ preserves orientation). Furthermore, ...
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### How to define addition law in hyperboloid model(lorentz space) of hyperbolic space

I know mobius addition and Einstein addition are well defined in Poincaré ball model . But how to define addition in hyperboloid model(lorentz space) of hyperbolic space,and can we define the exact ...
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### Existence of geodesically convex neighborhoods in semi-Riemannian manifolds.

I am studying the text by Barrett O’Neill referenced below. On page 130, O'Neill states, as Proposition 7, that every point in a semi-Riemannian Manifold has a convex neighborhood. Convex is defined ...