### Abstract

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

Original language | English |
---|---|

Pages (from-to) | 87-114 |

Number of pages | 28 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Jan |

Externally published | Yes |

### Fingerprint

### Keywords

- Bochner-weitzenböck formulas
- Casimir elements
- Invariant operators
- SO(n)-modules

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Bochner-weitzenböck formulas and curvature actions on Riemannian manifolds.** / Homma, Yasushi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Bochner-weitzenböck formulas and curvature actions on Riemannian manifolds

AU - Homma, Yasushi

PY - 2006/1

Y1 - 2006/1

N2 - Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

AB - Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

KW - Bochner-weitzenböck formulas

KW - Casimir elements

KW - Invariant operators

KW - SO(n)-modules

UR - http://www.scopus.com/inward/record.url?scp=30744455720&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30744455720&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-05-04068-7

DO - 10.1090/S0002-9947-05-04068-7

M3 - Article

AN - SCOPUS:30744455720

VL - 358

SP - 87

EP - 114

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -