I've been trying to grasp the concept of a geodesic sphere as put forward in Lee's Riemannian Manifolds. Here is some background.
Let $M$ be a smooth manifold, $TM$ be the tangent bundle, $\gamma_V$ be the unique maxial geodesic for any point $p \in M$ and $V\in T_pM$:
$\varepsilon = \{V \in TM$, where $V$ - an initial velocity vector, s.t. $\gamma_V$ is defined on some $I\in \mathbb{R}$, where $[0,1] \in I\}$
$exp: \varepsilon \to M$, such that $exp(V)=\gamma_V(1)$, and $exp_p: \varepsilon \cap T_pM \to M$.
A normal neighborhood of $p \in M$ is an open nbh $U$ that is the diffeomorphic image of a star-shaped open nbh of $0 \in T_pM$.
Given $\varepsilon>0$, such that $exp_p$ is a diffeomorphism on the closed ball $\bar{B}_\varepsilon(0)\in T_pM$, we call $exp_p(\bar{B}_\varepsilon(0))$ a closed geodesic ball. The notation for the geodesic sphere is $\partial exp_p(\bar{B}_\varepsilon(0))$. Now, I have two possible interpretations for the geodesic sphere and I believe both are incorrect.
First I thought it could perhaps be interpreted as the partial derivative of the function $exp_p$, evaluated on the set of vectors contained in $\bar{B}_\varepsilon (0)$. However, this brought me to the thought that the curvature on an abstract manifold would takes us off any hypothetical tangent space if we try to capture rate of change by moving along the surface of $M$.
The other thing I came up with is given any orthonormal basis $\{E_i\}$ for $T_pM$, the map $E: \mathbb{R}^n \to T_pM$ ,given by $E(x^1, \dots, x^n)=x^iE_i$ produces an isomorphism, and then combining maps: $\varphi:E^{-1}\circ exp_p^{-1}: U \to \mathbb{R}^n$ would give us the so called normal coordinates at $p$. So I thought that maybe as a subset of $M$, $exp_p(\bar{B}_\varepsilon(0))$ could be considered to have such a basis? But then again I have no idea how it relates to the concept of a sphere. I would appreciate any insights.