Positive-definite bilinear form in the context of Lorentzian manifold

⭐ Core Definition: Positive-definite bilinear form

In mathematics, a definite quadratic form is a quadratic form over some real vector space $V$ that has the same sign (always positive or always negative) for every non-zero vector of $V$ . According to that sign, the quadratic form is called positive-definite or negative-definite.

A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of $V$ .

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⭐ Core Definition: Positive-definite bilinear form
Positive-definite bilinear form in the context of Pseudo-Riemannian manifold
Positive-definite bilinear form in the context of Curvature invariant (general relativity)

Positive-definite bilinear form in the context of Pseudo-Riemannian manifold

In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.

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Positive-definite bilinear form in the context of Curvature invariant (general relativity)

In general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors – which represent curvature, hence the name – and possibly operations on them such as contraction, covariant differentiation and dualisation.

Certain invariants formed from these curvature tensors play an important role in classifying spacetimes. Invariants are actually less powerful for distinguishing locally non-isometric Lorentzian manifolds than they are for distinguishing Riemannian manifolds. This means that they are more limited in their applications than for manifolds endowed with a positive-definite metric tensor.

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