DEGENERATIONS OF LEVEL SETS OF THE MORSE FUNCTION ON THE UNIT TANGENT BUNDLE OF A SPHERE

Let $V_2\left(\mathbb{R}^{n+1}\right)$ be the Stiefel manifold of orthogonal 2-frames in $\mathbb{R}^{n+1}$. Fixing $c_1$ and $c_2$ which satisfy $0<c_1<c_2$, we define the well-known Morse function $f_{n+1,2}: V_2\left(\mathbb{R}^{n+1}\right) \rightarrow \mathbb{R}$. The function $f_{n+1,2}$ has four critical points, whose critical values are $\varepsilon_1 c_1+\varepsilon_2 c_2$, where $\varepsilon_1, \varepsilon_2 \in\{ \pm 1\}$. In this paper, we study the behavior of the degeneration maps $\zeta_1: f_{n+1,2}^{-1}(0) \rightarrow f_{n+1,2}^{-1}\left(-c_1+c_2\right)$ and $\zeta_2: f_{n+1,2}^{-1}(0) \rightarrow f_{n+1,2}^{-1}\left(c_1-c_2\right)$. Received: January 9, 2026Accepted: February 19, 2026