Let $M$ be a complete Riemannian manifold and $N$ be any compact submanifold of it. Let $\nu$ denotes the unit normal bundle of $N$. Let $p\in N$ and $x_0\in \nu_p = \{v\in T_pN^\perp: \|v\|=1\}$. Then prove that $\overline{B(x_0,\delta)}\cap \nu$ is homeomorphic to closed $(n-1)$ ball, where $n$ is the dimesnion of $M$ and $\overline{B(x_0,\delta)}$ is the closed ball with center $x_0$ and radius $\delta$.
Why I guessed it should be the ball of dimension $(n-1)$ as $\overline{B(x_0,\delta)}\cap \nu$ is the same as the intersection of a solid sphere with a cylinder (in $\mathbb{R}^3$) which is precisely a closed disk.
Edit
For example, consider the normal bundle $\mathbb{R}^k\times \{\mathbf{0}\}$. Then the unit normal bundle $\nu = \mathbb{R}^k \times \mathbb{S}^{n-k-1}$. Choose $\mathbf{x}_0 = (\mathbf{0},\mathbf{e}_1)\in \nu$. Then the ball $$B(\mathbf{x}_0,\delta) = \{(q,v)\in \mathbf{R}^k \times \mathbf{R}^{n-k}: \|q\|^2 + \|v-\mathbf{e}_1\|^2 \le \delta^2 \}$$. Now I claim that the set $$ \{(q,v)\in \mathbf{R}^k \times \mathbf{R}^{n-k}: \|q\|^2 + \|v-\mathbf{e}_1\|^2 \le \delta^2,~\|v\|=1 \} $$ is homeomorphic to $(n-1)$ closed ball.
Any hint or ideas will be appreciated. Thanks!