Invariant Riemannian Metrics Research Articles

Positively Curved Deformations of Invariant Riemannian Metrics

Let ${K_\gamma }$ denote the sectional curvature function of the Riemannian metric $\gamma$ on a manifold $M$. Suppose $M$ admits no metric $\gamma$ invariant under the action of a compact group $G$ and having ${K_\gamma } > 0$. It is shown that a $G$-invariant metric $\gamma (0)$ with ${K_{\gamma (0)}} \geqq 0$ cannot be embedded in a $1$-parameter family $\gamma (t)$ for which ${[d{K_{\gamma (t)}}/dt]_{t = 0}}$ is positive wherever ${K_{\gamma (0)}}$ is zero.

Remarks on unified field structures, spin structures and canonical quantization

Unified field structures are defined and reviewed. Under certain conditions these are shown to be dynamical systems. And quantizable dynamical systems are shown to be unified field structures with invariant Riemannian metric. Spin structure is reviewed and manifolds M 8k+4 with spin structure are shown to be symplectic.

Positively curved deformations of invariant Riemannian metrics

Let K γ {K_\gamma } denote the sectional curvature function of the Riemannian metric γ \gamma on a manifold M M . Suppose M M admits no metric γ \gamma invariant under the action of a compact group G G and having K γ > 0 {K_\gamma } > 0 . It is shown that a G G -invariant metric γ ( 0 ) \gamma (0) with K γ ( 0 ) ≧ 0 {K_{\gamma (0)}} \geqq 0 cannot be embedded in a 1 1 -parameter family γ ( t ) \gamma (t) for which [ d K γ ( t ) / d t ] t = 0 {[d{K_{\gamma (t)}}/dt]_{t = 0}} is positive wherever K γ ( 0 ) {K_{\gamma (0)}} is zero.

The automorphism group of a homogeneous almost complex manifold

1. Introduction. Let M be a compact simply connected manifold of nonzero Euler characteristic that carries a homogeneous almost complex structure. We determine the largest connected group A0(M) of almost analytic automorphisms of M. Our hypotheses represent M as a coset space G/K where G is a maximal compact subgroup of the Lie group A0(M) and Ais a closed connected subgroup of maximal rank in G. In §2 we collect some information, decomposing M=MX x ■ ■ ■ x Mt as a product of irreducible factors along the decomposition of G as a product of simple groups; then every invariant almost complex structure or riemannian metric decomposes and every invariant riemannian metric is hermitian relative to any invariant almost complex structure; furthermore the decomposition is independent of G in a certain sense. In §3 we choose an invariant riemannian metric and determine the largest connected groups Af0(Mi) of almost hermitian isometries of the Mi, Then A0(M) is determined in §4. There it is shown that A0(M) = A0(M1) x ■ ■ ■ xA0(Mt), that A0(Mi)=H0(Mi) if the almost complex structure on Mi is not integrable, and that AQ(Mi) = H0(Mi)c if the almost complex structure on M{ is induced by a complex structure. A0(M) thus is a centerless semisimple Lie group whose simple normal analytic subgroups are just the A0(Mi). 2. Decomposition. Let M be an effective coset space of a compact connected Lie group G by a connected subgroup A of maximal rank. In other words M=G/K is compact, simply connected and of nonzero Euler characteristic; G is a compact centerless semisimple Lie group, rank AJ=rank G, and K contains no simple factor of G. Then

A compatibility condition between invariant riemannian metrics and Levi-Whitehead decompositions on a coset space

0. Introduction. Let M = G/K be an effective coset space of a connected Lie group by a compact subgroup. Then there may be many G-invariant riemannian metrics on M. But one expects the algebraic structure of the pair (G, K) to have a strong influence on the curvatures of M relative to any G-invariant riemannian metric. For example (1) if G is semisimple with finite center and K is a maximal compact subgroup, then it is classical from symmetric space theory that all G-invariant riemannian metrics on M have every sectional curvature < 0; (2) if G is commutative then every G-invariant riemannian metric on M is flat; and (3) if G is noncommutative and nilpotent then [7] every G-invariant riemannian metric on M has sectional curvatures of both signs. Those results are proved by choosing an adG (K)-stable complement M to the Lie algebra A of K inside the Lie algebra 3 of G, and by performing calculations in 9O and in 3 in a manner justified by embedding G in the orthonormal frame bundle of M. But at certain crucial parts of those calculations one must have 3 either semisimple or nilpotent. The idea in this paper is to create a setup in which the calculations can still be carried out, by requiring that the complement 9O split as

Fibred spaces with invariant Riemannian metric

Fibred spaces with invariant Riemannian metric

On Holonomy and Homogeneous Spaces

In general a homogeneous space admits many invariant affine connections. Among these are certain connections which appear in many ways to be more natural than the others. We refer to the connections which K. Nomizu in [4] calls canonical affine connections of the first kind. When G is a compact connected Lie group and K a closed subgroup we called an invariant Riemannian metric on G/K, natural (in [2]) when it induced such a connection.