Einstein–Hermitian vector bundle

In differential geometry, an Einstein–Hermitian vector bundle is a Hermitian vector bundle over a Hermitian manifold whose metric is an Einstein–Hermitian metric, meaning that it satisfies the Einstein condition that the mean curvature, considered as an endomorphism of the vector bundle, is a constant times the identity operator. Einstein–Hermitian vector bundles were introduced by Kobayashi (1980, section 6). The Kobayashi–Hitchin correspondence implies that Einstein–Hermitian vector bundles are closely related to stable vector bundles. For example, every irreducible Einstein–Hermitian vector bundle over a compact Kähler manifold is stable.

See also

• Einstein manifold

References

• Kobayashi, Shoshichi (1980), "First Chern class and holomorphic tensor fields", Nagoya Mathematical Journal 77: 5–11, ISSN 0027-7630, http://projecteuclid.org/getRecord?id=euclid.nmj/1118786013 
• Kobayashi, Shoshichi (1987), Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, ISBN 978-0-691-08467-1