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| A \emph{pseudo-Riemannian} manifold is a manifold $M$ together with a non degenerate, symmetric section $g$ of $T^0_{2}(M)$ (2-covariant tensor bundle over $M$). |
A \emph{pseudo-Riemannian} manifold is a manifold $M$ together with a non degenerate, symmetric section $g$ of $T^0_{2}(M)$ (2-covariant tensor bundle over $M$). |
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| Unlike with a Riemannian manifold, $g$ is not positive definite. That is, there exist vectors $v\in T_{p}M$ such that $g(v,v)<0$. Non-degeneracy implies that $g(u,u)=0 \implies u=0$. |
Unlike with a Riemannian manifold, $g$ is not positive definite. That is, there exist vectors $v\in T_{p}M$ such that $g(v,v)<0$. Non-degeneracy implies that $g(u,u)=0 \implies u=0$. |
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| A well known \PMlinkname{result from linear algebra}{SylvestersLaw} permits us to make a change of basis such that in the new base $g$ is represented by a diagonal matrix with $-1$ or $1$ elements in the diagonal. If there are $i$, $-1$ elements in the diagonal and $j$, $1$, the tensor is said to have signature $(i,j)$ |
A well known \PMlinkname{result from linear algebra}{SylvestersLaw} permits us to make a change of basis such that in the new base $g$ is represented by a diagonal matrix with $-1$ or $1$ elements in the diagonal. If there are $i$, $-1$ elements in the diagonal and $j$, $1$, the tensor is said to have signature $(i,j)$ |
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| The signature will be invariant in every connected component of $M$, but usually the restriction that it be a global invariant is added to the definition of a pseudo-Riemannian manifold. |
The signature will be invariant in every connected component of $M$, but usually the restriction that it be a global invariant is added to the definition of a pseudo-Riemannian manifold. |
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| Unlike a Riemannian metric, some manifolds do not admit a pseudo-Riemannian metric. |
Unlike a Riemannian metric, some manifolds do not admit a pseudo-Riemannian metric. |
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| Pseudo-Riemannian manifolds are crucial in Physics and in particular in General Relativity where space-time is modeled as a 4-pseudo Riemannian manifold with signature (1,3)\footnote{also referred to as $(-+++)$}. |
Pseudo-Riemannian manifolds are crucial in Physics and in particular in General Relativity where space-time is modeled as a 4-pseudo Riemannian manifold with signature (1,3)\footnote{also referred to as $(-+++)$}. |
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| Intuitively pseudo-Riemannian manifolds are generalizations of Minkowski's space just as a Riemannian manifold is a generalization of a vector space with a positive definite metric. |
Intuitively pseudo-Riemannian manifolds are generalizations of Minkowski's space just as a Riemannian manifold is a generalization of a vector space with a positive definite metric. |
