By Lorenz J. Schwachhöfer (auth.), Christian Bär, Joachim Lohkamp, Matthias Schwarz (eds.)

This quantity includes a selection of well-written surveys supplied by means of specialists in worldwide Differential Geometry to offer an outline over contemporary advancements in Riemannian Geometry, Geometric research and Symplectic Geometry.

The papers are written for graduate scholars and researchers with a common curiosity in geometry, who are looking to get accustomed to the present traits in those important fields of recent mathematics.

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48, 1–42 (1926); ou Oeuvres compl`etes, tome III, vol. 2, 997–1038 34. : On the existence of infinite series of exotic holonomies. Inv. Math. a/ 35. : Exotic holonomies E7 . Int. Jour. Math. 8, 583–594 (1997) 36. : Alekseevskian spaces. Diff. Geom. Appl. 6, 129–168 (1996) 37. : Sur la r´eductibilit´e d’un espace de Riemann. Comm. Math. Helv. 26, 328–344 (1952) 38. : The geometry of homogeneous submanifolds in hyperbolic space, Math. Zeit. 237(1), 199–209 (2001) 39. DG/0406397 (2004) 40. : Metrics that realize all Lorentzian holonomy algebras.

4 Einstein Metrics on Four-Manifolds We now prove the following constraint on the topology of four-manifolds admitting Einstein metrics. Entropies, Volumes, and Einstein Metrics 47 Theorem 1. Let X be a closed oriented Einstein 4–manifold. X / the volume entropy. Equality in (5) occurs if and only if every Einstein metric on X is flat, is non-flat locally Calabi–Yau, or is of constant negative sectional curvature. Remark 1. X //, where f W X ! X / is the classifying map of the universal covering.

For this, we let G0 G be the connected subgroup with Lie algebra g0 Ä g. Since g0 Ä p and hence G0 P , it follows that we have a fibration P =G0 ! G=G0 ! v/ D 1g, where D T C denotes the contact distribution. , we have a canonical embedding { W Ca ,! G=G0 . Ca // G where W G ! G=G0 is the canonical projection. Then the restriction W a ! Ca / Š Ca becomes a principal G0 -bundle. G/ ˝ gi . Then i i iD 2 we can show the following. 1. C / and the principal G0 -bundle W a ! Ca with a G from above.