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is naturally identified with the vector space of
Killing vector fields, it follows that the isometry group is zero-dimensional. Bochner's theorem then follows from the fact that the isometry group of a closed Riemannian manifold is compact.
66:
cannot have a local maximum. In particular, on a closed
Riemannian manifold of negative Ricci curvature, every Killing vector field is identically zero. Since the isometry group of a complete Riemannian manifold is a
486:
has a local maximum, then it must be identically zero in a neighborhood. Since
Killing vector fields on connected manifolds are uniquely determined from their value and derivative at a single point, it follows that
358:{\displaystyle {\frac {1}{2}}\Delta \langle X,X\rangle =\langle \nabla X,\nabla X\rangle -\nabla _{X}\operatorname {div} X+\langle X,\operatorname {div} ({\mathcal {L}}_{X}g)\rangle -\operatorname {Ric} (X,X).}
577:
93:
378:
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The theorem is a corollary of
Bochner's more fundamental result which says that on any connected Riemannian manifold of negative Ricci curvature, the length of a nonzero
1179:
896:
817:
1198:
741:{\displaystyle \nabla ^{p}\nabla _{p}X_{i}=-\nabla _{i}\nabla ^{p}X_{p}+\nabla ^{p}(\nabla _{i}X_{p}+\nabla _{p}X_{i})-R_{ip}X^{p}.}
1172:
809:
17:
186:{\displaystyle \Delta X=-\nabla (\operatorname {div} X)+\operatorname {div} ({\mathcal {L}}_{X}g)-\operatorname {Ric} (X,\cdot )}
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1108:. Classical Topics in Mathematics. Vol. 6 (New expanded ed.). Beijing: Higher Education Press. pp. 30–32.
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464:{\displaystyle {\frac {1}{2}}\Delta \langle X,X\rangle =\langle \nabla X,\nabla X\rangle -\operatorname {Ric} (X,X).}
1153:
478:. However, on a Riemannian metric of negative Ricci curvature, the right-hand side is strictly positive wherever
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1008:
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In the case of a
Riemannian metric, the left-hand side is nonpositive at any local maximum of the length of
858:
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930:. Interscience Tracts in Pure and Applied Mathematics. Vol. 15. Reprinted in 1996. New York–London:
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1061:. Applied Mathematical Sciences. Vol. 116 (Second edition of 1996 original ed.). New York:
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Isometry group of a compact
Riemannian manifold with negative Ricci curvature is finite
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Partial differential equations II. Qualitative studies of linear equations
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Boothby, William M. (1954). "Book Review: Curvature and Betti numbers".
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Bochner's result on
Killing vector fields is an application of the
976:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 70.
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on a pseudo-Riemannian manifold. As a consequence, there is
1011:. Vol. 171 (Third edition of 1998 original ed.).
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867:. Annals of Mathematics Studies. Vol. 32.
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1106:The Bochner technique in differential geometry
970:Transformation groups in differential geometry
967:(1972). "Isometries of Riemannian Manifolds".
372:is a Killing vector field, this simplifies to
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818:Bulletin of the American Mathematical Society
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574:In an alternative notation, this says that
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83:as follows. As an application of the
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54:of the manifold must be finite.
910:10.1090/S0002-9904-1954-09834-8
831:10.1090/S0002-9904-1946-08647-4
50:must be zero. Consequently the
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1152:. You can help Knowledge by
85:Ricci commutation identities
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1204:Differential geometry stubs
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864:Curvature and Betti numbers
781:Kobayashi & Nomizu 1963
563:Kobayashi & Nomizu 1963
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196:holds for any vector field
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491:must be identically zero.
1071:10.1007/978-1-4419-7052-7
1021:10.1007/978-3-319-26654-1
1003:Petersen, Peter (2016).
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38:proved in 1946 that any
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918:Kobayashi, Shoshichi
771:, Proposition 8.2.1.
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509:, Corollary VI.5.4;
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64:Killing vector field
40:Killing vector field
18:Bochner–Yano theorem
1005:Riemannian geometry
44:Riemannian manifold
1055:Taylor, Michael E.
980:. pp. 55−57.
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549:, Theorem VI.3.4;
513:, Corollary 8.2.3.
482:is nonzero. So if
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1080:978-1-4419-7051-0
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1193:Categories
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878:0691095833
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798:References
58:Discussion
710:−
688:∇
665:∇
652:∇
629:∇
619:∇
615:−
593:∇
583:∇
441:
435:−
432:⟩
426:∇
417:∇
414:⟨
408:⟩
396:⟨
393:Δ
335:
329:−
326:⟩
300:
288:⟨
279:
267:∇
263:−
260:⟩
254:∇
245:∇
242:⟨
236:⟩
224:⟨
221:Δ
178:⋅
166:
160:−
134:
119:
110:∇
107:−
98:Δ
69:Lie group
1063:Springer
1057:(2011).
924:(1963).
861:(1953).
808:(1946).
1124:3838345
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438:Ric
332:Ric
297:div
276:div
163:Ric
131:div
116:div
30:In
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