Mean curvature vector

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $f:M^m\to (\bar{M},\bar{g})$ be a smooth immersion of positive dimension $m\ge 1$, and give $M$ the induced metric $g=f^{\ast }\bar{g}$. The orthogonal complements of $df(TM)$ in $f^{\ast }T\bar{M}$ form the smooth normal bundle ${\nu }_{f}M$. Every immersion is locally an embedding, and on such a neighbourhood the embedded second fundamental form from The second fundamental form is a symmetric normal-bundle-valued two-tensor pulls back to a section of $S^2{T}^{\ast }M\otimes {\nu }_{f}M$. Equivalently it is the normal projection of $(f^{\ast }\bar{\mathrm{\nabla }}{)}_{X}df(Y)$. This intrinsic pullback- connection formula shows that the local tensors agree on overlaps; denote the result by ${\mathrm{I}\mathrm{I}}_f$.

The immersion's mean curvature vector field (with the averaged convention) is

${\mathbf{H}}_f:=\frac{1}{m}{\mathrm{tr}}_g{\mathrm{I}\mathrm{I}}_f\in \mathrm{\Gamma }({\nu }_{f}M)\mathrm{.}$

Thus, at $p\in M$, for any orthonormal basis $(e_1,\dots ,e_m)$ of $T_{p}M$,

${\mathbf{H}}_f(p)=\frac{1}{m}{\sum }_{i=1}^m({\mathrm{I}\mathrm{I}}_f{)}_p(e_i,e_i)\mathrm{.}$

This value is independent of the orthonormal basis. Indeed, if $f_a={\sum }_i{O}_{ai}{e}_i$ is another one, then $O$ is orthogonal, and bilinearity of the normal-bundle-valued tensor ${\mathrm{I}\mathrm{I}}_f$ gives

${\sum }_a{\mathrm{I}\mathrm{I}}_f(f_a,f_a)={\sum }_{i,j}\left({\sum }_a{O}_{ai}{O}_{aj}\right){\mathrm{I}\mathrm{I}}_f(e_i,e_j)={\sum }_i{\mathrm{I}\mathrm{I}}_f(e_i,e_i)\mathrm{.}$

Equivalently, this is contraction of the two covariant tangent slots after raising one of them with the inverse metric. In a smooth local tangent frame $(X_i)$ it has the formula

${\mathbf{H}}_f=\frac{1}{m}{\sum }_{i,j}{g}^{ij}{\mathrm{I}\mathrm{I}}_f(X_i,X_j)\mathrm{.}$

The inverse-metric coefficients $g^{ij}$ and the coefficients of ${\mathrm{I}\mathrm{I}}_f$ are smooth, so this formula also proves that ${\mathbf{H}}_f$ is a smooth normal field. Componentwise, its invariance is the usual basis-independence of contraction.

No normal frame, normal orientation, or coorientation enters the definition, so ${\mathbf{H}}_f$ is independent of all such choices. For an embedded submanifold and its inclusion, this is exactly the preceding embedded construction. Calegari's Warning 2.4 calls the displayed averaged value the more usual convention; Calegari and Terng use the unnormalized trace instead. Consequently, their mean-curvature vector is $m{\mathbf{H}}_f$ in the present notation, which accounts for the factor $m$ in the next first-variation formula.

The assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited exactly through the smooth normal projection used to construct ${\mathrm{I}\mathrm{I}}_f$; taking this finite trace adds no choice. The definition is the unique empty normal field when $M$ is empty of fixed positive dimension. In dimension one it is ${\mathbf{H}}_f(p)=({\mathrm{I}\mathrm{I}}_f{)}_p(e,e)$ for either unit tangent vector $e$; in codimension zero it is zero. It applies unchanged at boundary points. Dimension zero is excluded because $1\mathrm{/}m$ is undefined, and degenerate metrics are outside the Riemannian hypothesis.