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To get a greater suppose for this functionality think about the vector box X box twin to i. e. , for all v. Then Proposition 9. 2. 1. If X and are similar by way of , then (1) v ↦ ∇ v X is symmetric if and provided that and (2) . evidence. bear in mind that due to the fact L X g is symmetric and is skew-symmetric the outcome instantly follows. the second one half was once confirmed in proposition 2. 2. 7. □  for that reason, while is harmonic, then and ∇X is a symmetric -tensor. We current the Bochner formulation for closed 1-forms formulated via vector fields. Proposition 9. 2. 2. permit X be a vector box in order that ∇X is symmetric (i. e. corresponding 1-form is closed). If and X is the gradient of u close to p, then (1) (2) (3) facts. For (1) easily realize that (2) is an instantaneous end result of theorem 3. 2. 2 utilized to the functionality u. For (3) take strains in (2). As within the facts of proposition 8. 2. 1 this offers us the 1st and 3rd phrases. the second one time period comes from commuting strains and covariant derivatives. particularly, both X |  p  = 0 or E i will be able to be selected to parallel alongside X. In both case  □  nine. 2. 2 The Vanishing Theorem we will be able to now simply determine the opposite Bochner theorem for 1-forms. Theorem 9. 2. three (Bochner, 1948). If (M,g) is compact and has , then each harmonic 1-form is parallel. facts. consider ω is a harmonic 1-form and X the twin vector box. Then proposition 9. 2. 2 implies considering that . the utmost precept then indicates that needs to be consistent and . □  Corollary 9. 2. four. If (M,g) is as prior to and in addition has confident Ricci curvature at one element, then all harmonic 1-forms vanish all over. evidence. considering we simply proved we should have that X |  p  = 0 if the Ricci tensor is optimistic on T p M. yet then on the grounds that X is parallel. □  Corollary 9. 2. five. If (M,g) is compact and satisfies , then b 1 (M) ≤ n = dim M, with equality keeping if and provided that (M,g) is a flat torus. facts. we all know from Hodge idea that . Now, all harmonic 1-forms are parallel, so the linear map: that evaluates ω at p is injective. specifically, . If equality holds, then there are n linearly autonomous parallel fields E i ,  i = 1, …, n. This in actual fact signifies that (M, g) is flat. therefore the common overlaying is with performing by way of isometries. Now pull the vector fields E i ,  i = 1, …, n, back to on . those vector fields are back parallel and accordingly consistent vector fields. which means we will examine them because the traditional Cartesian coordinate vector fields ∂ i . moreover, they're invariant lower than the motion of , i. e. , for every we have now i = 1, …, n. yet in simple terms translations go away all the coordinate fields invariant. hence, is composed solely of translations. which means is finitely generated, Abelian, and torsion loose. as a result for a few okay. to determine that M is a torus, we want k = n. If k < n, then generates a formal subspace of the gap of translations and can’t act cocompactly on . If k > n, then can’t act discretely on . therefore, it follows that and generates  □  nine. 2. three The Estimation Theorem The target is to generalize theorem 9. 2. three to manifolds with a destructive decrease sure for the Ricci curvature.