The Newman–Penrose formalism for Riemannian 3-manifolds

Abstract

Abstract. We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field k with geodesic flow, if an integral curve of k is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if k is complete, hypersurfaceorthogonal, and satisfies Ric(k, k) ≥ 0, then its divergence must be nonnegative. As an application, we show that if the Riemannian 3-manifold is closed and a unit length k with geodesic flow satisfies Ric(k, k) > 0, then k cannot be hypersurface-orthogonal, thus recovering a result in [7]. Turning next to scalar curvature, we derive an evolution equation for the scalar curvature in terms of unit vector fields k that satisfy the condition R(k, ·, ·, ·) = 0. When the scalar curvature is a nonzero constant, we show that a hypersurface-orthogonal unit vector field k satisfies R(k, ·, ·, ·) = 0 if and only if it is a Killing vector field.

Symplectic 4-manifolds via Lorentzian geometry

“…In Section 2, we will provide an example of a symplectic form, constructed as in Theorem 1, on the Lorentzian 4-manifold (R × S 3 , −dt 2 ⊕g), whereg is the round metric on the 3-sphere S 3 ; this symplectic form, it turns out, will be the symplectization of a contact form on S 3 whose general form was first discovered in [12]. Specifically, [12] showed that if k is a unit vector field on a Riemannian 3-manifold (M 3 , g) satisfying ∇ k k = 0 and Ric(k, k) > 0, then the 1-form g(k, ·) is a contact form; this was then re-derived, by another means, in [1]. Theorem 1 above is essentially a fourdimensional symplectic version of the construction in [1], made possible for the following reason: because a null vector field k uniquely satisfies k ⊂ k ⊥ , one can thus consider the two-dimensional quotient subbundle k ⊥ /k instead of the full three-dimensional subbundle k ⊥ -this is the crucial (and well known) fact that ultimately makes Theorem 1 possible.…”

“…Together with a function f such that k(f ) is nowhere vanishing, this is enough to ensure that the closed 2-form dg(e f k, ·) is nondegenerate. In Section 2 we also provide two examples to show that: (1) positive Ricci curvature is a sufficient, but by no means necessary, condition to ensure twistedness; (2) the assumption of the completeness of k cannot be dropped from Theorem 1.…”

“…Specifically, [12] showed that if k is a unit vector field on a Riemannian 3-manifold (M 3 , g) satisfying ∇ k k = 0 and Ric(k, k) > 0, then the 1-form g(k, ·) is a contact form; this was then re-derived, by another means, in [1]. Theorem 1 above is essentially a fourdimensional symplectic version of the construction in [1], made possible for the following reason: because a null vector field k uniquely satisfies k ⊂ k ⊥ , one can thus consider the two-dimensional quotient subbundle k ⊥ /k instead of the full three-dimensional subbundle k ⊥ -this is the crucial (and well known) fact that ultimately makes Theorem 1 possible. Regarding the existence of the function f , there is a well known class of Lorentzian 4-manifolds, namely, the globally hyperbolic ones, which possess Cauchy temporal functions f as defined in [3,14], which naturally satisfy the property that k(f ) is nowhere vanishing.…”

Ricci-positive geodesic flows and point-completion of static monopole fields

“…as in recent work of Aazami [4], where the Newman-Penrose formalism has been applied with interesting effect to methods of symplectic geometry, then the Kähler structure may be recovered from a simple reversal of orientation onM , effectively replacing j by −j. The presence of a Kähler structure on π −1 (M \K) enables the standard presentation of the Bogomolny equation as a time-independent reduction of the anti self-dual property of the connection…”

“…as in recent work of Aazami [4], where the Newman-Penrose formalism has been applied with interesting effect to methods of symplectic geometry, then the Kähler structure may be recovered from a simple reversal of orientation on M , effectively replacing j by −j.…”