Ngô Quốc Anh

October 11, 2013

Bochner-type formula for the conformal Killing operator on manifolds without boundary

Filed under: Uncategorized — Tags: , — Ngô Quốc Anh @ 9:59

Assuming the Riemannian manifold (M,g) is compact without boundary. In the previous post, we showed that

\displaystyle \frac{1}{2}\int_M |\mathbb L_X g|^2 dv_g= \int_M |\nabla X|^2 dv_g +\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g .

Today, we use the above formula to derive a Bochner-type formula for the conformal Killing operator \mathbb L given by

\displaystyle \mathbb L X = \mathbb L_Xg - \frac{2}{n}\text{div}(X)g.

Clearly, for any vector field X,

\begin{array}{lcl} |\mathbb{L}X{|^2} &=& \displaystyle \Big|{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g \Big|^2 \hfill \\ &=&\displaystyle {\left( {{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g} \right)_{im}}{\left( {{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g} \right)_{jn}}{g^{ij}}{g^{mn}} \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{{{n^2}}}{( \text{div}X)^2}\underbrace {{g_{im}}{g_{jn}}{g^{ij}}{g^{mn}}}_{\delta _n^i\delta _i^n = n} - \frac{2}{n}( \text{div}X)\left[ {{g_{im}}{\mathbb{L}_X}{g_{jn}} + {g_{jn}}{\mathbb{L}_X}{g_{im}}} \right]{g^{ij}}{g^{mn}} \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[({\mathbb{L}_X}{g_{jn}})\underbrace {{g_{im}}{g^{ij}}{g^{mn}}}_{\delta _m^j{g^{mn}} = {g^{jn}}} + ({\mathbb{L}_X}{g_{im}})\underbrace {{g_{jn}}{g^{ij}}{g^{mn}}}_{\delta _n^i{g^{mn}} = {g^{im}}}] \hfill \\ &=&\displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[({\nabla _j}{X_n}+\nabla_n X_j){g^{jn}} + ({\nabla _i}{X_m}+\nabla_m X_i){g^{im}}] \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[\underbrace {2{\nabla ^n}{X_n} + 2{\nabla ^m}{X_m}}_{4 \text{div}X}] \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} - \frac{4}{n}{( \text{div}X)^2}. \end{array}

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