Induced connection and second fundamental form

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M\subseteq \bar{M}$ be an embedded Riemannian submanifold, let $\bar{\mathrm{\nabla }}$ be the Levi–Civita connection of the ambient metric, and let $X,Y\in \mathrm{\Gamma }(TM)$. Around each $p\in M$, extend $X$ and $Y$ locally to ambient fields $\tilde{X}$ and $\tilde{Y}$ using a slice chart, and define the ambient derivative along $M$ by

${\bar{\mathrm{\nabla }}}_{X}Y{\mathrm{\mid }}_p:=({\bar{\mathrm{\nabla }}}_{\tilde{X}}\tilde{Y}{)}_p\mathrm{.}$

This value is independent of both extensions. Indeed, an alternative first extension differs at $p$ by a vector that is zero, so function-linearity in the differentiating slot gives no change. If an alternative second extension differs by $W={\sum }_a{f}^a{\mathrm{\partial }}_a$ with $W{\mathrm{\mid }}_M=0$, then every $f^a{\mathrm{\mid }}_M=0$ and tangency of $X_p$ gives $X_p(f^a)=0$. The connection laws therefore give

$({\bar{\mathrm{\nabla }}}_{\tilde{X}}W{)}_p={\sum }_a{X}_p(f^a){\mathrm{\partial }}_a{\mathrm{\mid }}_p+{\sum }_a{f}^a(p)({\bar{\mathrm{\nabla }}}_{\tilde{X}}{\mathrm{\partial }}_a{)}_p=0.$

The same local calculation shows that these values vary smoothly along $M$. No simultaneous or global choice of extensions is used.

Using the smooth projections of Tangential and normal projections along a Riemannian submanifold, define the induced connection and the second fundamental form by

${\mathrm{\nabla }}_X^{M}Y:=({\bar{\mathrm{\nabla }}}_{X}Y{)}^{\mathrm{\top }},\mathrm{I}\mathrm{I}(X,Y):=({\bar{\mathrm{\nabla }}}_{X}Y{)}^{\perp }\mathrm{.}$

Thus the orthogonal splitting gives the Gauss decomposition

${\bar{\mathrm{\nabla }}}_{X}Y={\mathrm{\nabla }}_X^{M}Y+\mathrm{I}\mathrm{I}(X,Y)\mathrm{.}$

The assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited exactly from the preceding smooth restricted-bundle and projection construction. The local extension and independence calculation above adds no choice. The formulas are valid for the empty submanifold, in tangent or normal rank zero, in rank one, and at boundary points; positive definiteness of the Riemannian metric excludes a degenerate orthogonal splitting. The following items prove that ${\mathrm{\nabla }}^M$ is the intrinsic Levi–Civita connection and that $\mathrm{I}\mathrm{I}$ is a symmetric $\nu M$-valued tensor.