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Results tagged with riemannian-geometry
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user 259525
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
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Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there a …
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6
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On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (co …
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On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one rec …
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2
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Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?
Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that
$$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$
where $\Sigma …
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6
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answer
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Existence of adjoint operators on manifolds
Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise …
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Etymology “Kulkarni–Nomizu product”
$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two symme …
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