Bochner identity

The Bochner identity is the general name for a family of identities in Riemannian geometry that connect Laplacians of different types and curvature . The identities obtained by integrating the Bochner identities are sometimes called Reilly identities .

Content

1 Formulation
- 1.1 Designations
2 Consequences
3 notes
4 Literature

Wording

Let be $S$ ${\ displaystyle S}$ there is a Dirac bundle over a Riemannian manifold $M$ ${\ displaystyle M}$ , $D$ ${\ displaystyle D}$ Is the corresponding Dirac operator , and then

D^{2}\phi =\nabla ^{*}\nabla \phi +{\mathfrak {R}}(\phi )

{\ displaystyle D ^ {2} \ phi = \ nabla ^ {*} \ nabla \ phi + {\ mathfrak {R}} (\ phi)}

for any section $\phi \colon M\to S$ ${\ displaystyle \ phi \ colon M \ to S}$ .

Conventions

Further $e_{i}$ ${\ displaystyle e_ {i}}$ denotes an orthonormal frame at a point.

$\nabla$ ${\ displaystyle \ nabla}$ denotes connectivity on $S$ ${\ displaystyle S}$ , and
$\nabla ^{*}\nabla \phi =-\sum _{i}\nabla _{e_{i}}\nabla _{e_{i}}\phi ,$ ${\ displaystyle \ nabla ^ {*} \ nabla \ phi = - \ sum _ {i} \ nabla _ {e_ {i}} \ nabla _ {e_ {i}} \ phi,}$

the so-called Laplacian in connection .

${\mathfrak {R}}$ ${\ displaystyle {\ mathfrak {R}}}$ - section $\mathrm {Hom} (S,S)$ ${\ displaystyle \ mathrm {Hom} (S, S)}$ defined as
${\mathfrak {R}}(\phi )={\tfrac {1}{2}}\cdot \sum _{i,j}e_{i}.e_{j}.R_{e_{i},e_{k}}\phi ,$ ${\ displaystyle {\ mathfrak {R}} (\ phi) = {\ tfrac {1} {2}} \ cdot \ sum _ {i, j} e_ {i} .e_ {j} .R_ {e_ {i }, e_ {k}} \ phi,}$

where "

.

{\ displaystyle.}

" Stands for Clifford's multiplication , and

R_{X,Y}=\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{[X,Y]}

{\ displaystyle R_ {X, Y} = \ nabla _ {X} \ nabla _ {Y} - \ nabla _ {Y} \ nabla _ {X} - \ nabla _ {[X, Y]}}

- transformation of curvature .

$D$ ${\ displaystyle D}$ - Dirac operator on $S$ ${\ displaystyle S}$ , i.e
$D\phi =\sum _{i}e_{i}.\nabla _{e_{i}}\phi .$ ${\ displaystyle D \ phi = \ sum _ {i} e_ {i}. \ nabla _ {e_ {i}} \ phi.}$

and

D^{2}\phi =\Delta \phi

{\ displaystyle D ^ {2} \ phi = \ Delta \ phi}

Hodge Laplacian on differential forms

Consequences

From the Bochner identity for a gradient function $u$ ${\ displaystyle u}$ we obtain the following integral formula for any closed manifold
$\int \limits _{M}|\Delta u|^{2}-\int \limits _{M}|\mathrm {Hess} \,u|^{2}=\int \limits _{M}{\mbox{Ric}}(\nabla u,\nabla u)$ ${\ displaystyle \ int \ limits _ {M} | \ Delta u | ^ {2} - \ int \ limits _ {M} | \ mathrm {Hess} \, u | ^ {2} = \ int \ limits _ { M} {\ mbox {Ric}} (\ nabla u, \ nabla u)}$ ,

Where

\mathrm {Hess} \,u

{\ displaystyle \ mathrm {Hess} \, u}

denotes hessian

u

{\ displaystyle u}

.

If $u\colon M\rightarrow \mathbb {R}$ ${\ displaystyle u \ colon M \ rightarrow \ mathbb {R}}$ Is the harmonic function then
$\Delta ({\tfrac {1}{2}}\cdot |\nabla u|^{2})=|\nabla ^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u)$ ${\ displaystyle \ Delta ({\ tfrac {1} {2}} \ cdot | \ nabla u | ^ {2}) = | \ nabla ^ {2} u | ^ {2} + {\ mbox {Ric}} (\ nabla u, \ nabla u)}$ ,

Where

\nabla u

{\ displaystyle \ nabla u}

denotes a gradient

u

{\ displaystyle u}

. In particular:

Compact manifolds with positive Ricci curvature do not admit nonzero harmonic functions.
If $u$ ${\ displaystyle u}$ Is a harmonic function on a manifold with positive Ricci curvature, then the function $|\nabla u|$ ${\ displaystyle | \ nabla u |}$ subharmonic .

It follows from the Bochner formula that on compact manifolds with a positive curvature operator there are no harmonic forms of any degree, that is, it is a rationally homological sphere.
- By another method, namely the Ricci flow , it was possible to prove that any such manifold is diffeomorphic to the sphere factor with respect to a finite group. ^[one]

Notes

↑ B. Wilking, C. Böhm. Manifolds with positive curvature operators are space forms (English) // Ann. of Math. (2). - 2008 .-- Vol. 167 , no. 3 . - P. 1079-1097 .

Literature

H. Blaine Lawson, Marie-Louise Michelsohn. Spin geometry. - 1989.

[1] B. Wilking, C. Böhm. Manifolds with positive curvature operators are space forms (English) // Ann. of Math. (2). - 2008 .-- Vol. 167 , no. 3 . - P. 1079-1097 .