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        "quote": "or equivalently, $g ( X , X ) \\geq 0$ for all vector fields $X \\in \\mathcal { T } ( M )$ . In this case all $\\eta _ { i } = 1$ in the above expansion. The word Riemannian is also applied to the negative definite case, all $\\eta _ { i } = - 1$ . If the inner product defined by $g _ { p }$ on every tangent space is <u class=\"concept-term\" data-concept-identity=\"urn:uuid:2061e90b-c48d-463e-8b85-b8e1c592f56f\" data-concept-route=\"/concept-identities#concept-2061e90b-c48d-463e-8b85-b8e1c592f56f\" id=\"first-20d63bae569b760a93d1-en\">Minkowskian</u>, as defined in <a href=\"../chapter-05/01-real-inner-product-spaces#section-5-1\" class=\"cross-reference\" id=\"xref-04-pseudo-riemannian-manifolds-2\" data-reference=\"section-5-1\">Section 5.1</a>, we say $( M , g )$ is a Minkowskian or <u class=\"concept-term\" data-concept-identity=\"urn:uuid:6fd2c3b2-de81-4166-9379-10cec9081f7d\" data-concept-route=\"/concept-identities#concept-6fd2c3b2-de81-4166-9379-10cec9081f7d\" id=\"first-35631418c6c79a278bc2-en\">hyperbolic manifold</u>. In this case there exists a local orthonormal set of vector fields $e _ { i }$ such that the associated coefficients are $\\eta _ { 1 } = \\epsilon , \\eta _ { 2 } = \\cdot \\cdot \\cdot = \\eta _ { n } = - \\epsilon$ where $\\epsilon = \\pm 1$",
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