We start our discussion by introducing the basic notions of the normal exponential map and normal coordinates with respect to an embedded submanifold. This part seems to be standard in Riemannian geometry. For the consistence of the notations and the completeness of the paper, here we include detailed discussions and proofs.

Let $(Q,g)$ be an oriented Riemannian manifold of dimension $m$ and let $P\subset Q$ be a closed embedded oriented submanifold of codimension $k_{0}$ in $Q$. In our later applications, we only need the case $k_{0}=3$. Denote by ${N}$ the normal bundle of $P$ in $Q$, equipped with the induced fiberwise Riemannian inner products. For any $p\in P$, $N(p)$ denotes the fiber of $p$ in $N$. The normal exponential map of $P$ in $Q$ is defined as

where $\Exp_{p}:T_{p}Q\to Q$ is the standard exponential map at $p\in Q$. By standard implicit function theorem, it is straightforward that $\Exp_{P}:{N}\to Q$ is a diffeomorphism from some neighborhood of the zero section of ${N}$ to some tubular neighborhood of $P$ in $Q$.

Now we define the normal coordinates. Fix $p\in P$. First we choose local coordinates $\{x_{1}^{\prime},\ldots,x_{m-k_{0}}^{\prime}\}$ in a small neighborhood $U\subset P$ of $p$ such that

at $p$. We also assume that $\partial_{x_{1}^{\prime}}\wedge\cdots\wedge\partial_{x_{m-k_{0}}^{\prime}}$ is compatible with the orientation on $U$. Next, we pick local orthonormal sections $\{e_{1},\ldots,e_{k_{0}}\}$ of the normal bundle $N$ such that

on $U$. Again we assume that $e_{1}\wedge\cdots\wedge e_{k_{0}}$ is compatible with the orientation on $N$, i.e. $e_{1}\wedge\cdots\wedge e_{k_{0}}\wedge\partial_{x_{1}^{\prime}}\wedge\cdots\partial_{x_{k-m_{0}}^{\prime}}$ is compatible with the orientation on $Q$. Then we can find $\epsilon>0$ such that $\Exp_{p}$ restricts to a diffeomorphism from

to a neighborhood $\mathcal{U}$ of $p$. In particular, $U=\mathcal{U}\cap P$.

By definition for each fixed point $(y_{1},\ldots,y_{k_{0}},x_{1},\ldots,x_{m-k_{0}})$ in $\mathcal{U}$, the curve

is a normal geodesic which is orthogonal to $U\subset P$. Let

be the induced coordinate vector fields. Then $\partial_{y_{\alpha}}|_{y=0}=e_{\alpha}$ when both are viewed as sections of $N$ over $U\subset P$. For $1\leq i,j\leq m-k_{0}$ and $1\leq\alpha\leq k_{0}$, we denote

By definition, we have

for all $x\in U\subset P$. The second fundamental form of the embedding $U\hookrightarrow\mathcal{U}$ can be written as $\IIs=\IIs_{ij}^{\alpha}\partial_{y_{\alpha}}\otimes(dx_{i}\otimes dx_{j})$, where

Denote by $\overrightarrow{H}=H^{\alpha}e_{\alpha}$ the mean curvature vector, then

In the above coordinates, we define the normal distance function

A straightforward extension of the usual Gauss Lemma gives the following and we omit the proof.

For the convenience of later discussion, we introduce several notations concerning the normal regularity order near the submanifold $P$. It will be frequently used throughout the paper.

Notice that the defining condition does not depend on the choice of the local coordinates, since if we have another coordinate system $\{\tilde{y}_{1},\cdots,\tilde{y}_{k_{0}},\tilde{x}_{1},\cdots,\tilde{x}_{m-k_{0}}\}$, then we have

and $\frac{\partial y_{\alpha}}{\partial\tilde{x}_{j}}\in rO^{\prime}(1)$. Similarly, we can also use any local coordinate system $\{y_{\alpha},x_{i}\}$ such that $y_{\alpha}=0$ along $P$ for $1\leq\alpha\leq k_{0}$.

As a digression we briefly discuss the intrinsic meaning of the above expansion. The point is that locally the Riemannian metric $g$ is approximated by a Riemannian metric $g_{N}$ on the normal bundle $N$ up to the first order. Notice we have the natural projection $\pi:N\rightarrow P$ and $N$ is a Riemannian vector bundle together with an induced “normal” connection, given by the normal component of the Levi-Civita connection of $g$. The latter hence gives rise to a distribution of horizontal subspaces at each point of $N$, which in our coordinates is spanned by $\partial_{x_{i}}-A_{i\alpha\beta}y_{\beta}\partial_{y_{\alpha}}$, where

is a smooth function on $U$. We define $g_{N}$ so that at each point of $N$, the vertical and horizontal subspaces are orthogonal and on the vertical part is given by the bundle metric on $N$, and on the horizontal part is given by the perturbation of the base metric $g_{P}$ using the second fundamental form. In this way we get a coordinate free description of the above expansion up to the first order.

We also define

Then the curvature of the normal connection is given by