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Contact homogeneity and envelopes of Riemannian metrics. (English) Zbl 0892.53010

The authors define two Riemannian manifolds \((M,g)\) and \((\overline M,\overline g)\) to have contact of order \(s\) at the points \(p\in M\) and \(\overline p\in\overline M\) if there exist coordinates around \(p\) and \(\overline p\) such that the metric components and all their partial derivatives up to order \(s\) are the same at \(p\) and \(\overline p\). The notion of contact homogeneity is closely related to that of curvature homogeneity. Recall that a Riemannian manifold \((M,g)\) is curvature homogeneous up to order \(s\) if for every pair of points \(p,q\in M\) there exists a linear isometry \(\varphi\colon T_pM \to T_qM\) preserving the curvature tensor \(R\) and its covariant derivatives up to order \(s\), i.e., \(\varphi^*\nabla^i R_q=\nabla^i R_p\) for \(i=0,1,\dots,s\).
The authors show that a Riemannian manifold is curvature homogeneous up to order \(s\) if and only if it is contact homogeneous up to order \(s+2\). Hence, the problems and results in the field of curvature homogeneity can all be rephrased in the more basic language of contact homogeneity. (See E. Boeckx, O. Kowalski and L. Vanhecke [“Riemannian manifolds of conullity two” (World Scientific, River Edge, NJ) (1996)] for an overview of this topic.)
Next, let \(M\) be a manifold and \(\{g^p\mid p\in M\}\) a family of Riemannian metric on neighborhoods \(U_p\) of the corresponding points \(p\in M\). A Riemannian metric \(g\) on \(M\) is called a \(k\)-th order envelope of the family \(\{g^p\}\) if for each \(p\in M\), \((M,g)\) and \((U_p,g^p)\) have contact of order \(k\) at \(p\). Of course, every Riemannian metric on \(M\) is a first order envelope of (locally) Euclidean metrics, a fact proved explicitly here in a given coordinate system. Further, it is shown that every semi-symmetric metric \(g\) on \(M\) (i.e., such that the curvature tensor \(R\) satisfies \(R(X,Y)\cdot R=0\) for all vector fields \(X\), \(Y\) on \(M\)) is a second order envelope of locally symmetric metrics. This result had already been announced (without precise statement) by O. Kowalski [Czech. Math. J. 46, 427-474 (1996; Zbl 0879.53014)].

MSC:

53B20 Local Riemannian geometry
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds

Citations:

Zbl 0879.53014
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