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You can get a large class of examples if you look at the real projective space $\mathbb{RP}^n$. Each point $L\in \mathbb{RP}^n$ is a one dimensional subspace in $\mathbb{R}^{n+1}$ and we denote by $P_L$ the orthogonal projection onto $L$. Fix a unit vector $v\in \mathbb{R}^{n+1}$ and denote by $U$ the orthogonal complement in $\mathbb{R}^{n+1}$ of the line $L_v$ spanned by $v$. Consider the function

$$f_v: \mathbb{RP}^n\to\mathbb{R}, \;\; f_v(L)=(P_Lv,v).$$

Note that

$$0 \leq f_v(L)\leq 1,\;\;\forall L.$$

This is a Morse-Bott function with precisely two critical submanifolds: the locus of minima where $f=0$ and consisting of the real projective space $\mathbb{P}(U)\subset \mathbb{RP}^n$ consisting of lines in $U$, and a unique maximum point, $L_v\in\mathbb{RP}^n$ where $f_v(L_v)=1$.

The normal bundle of $\mathbb{P}(U)\subset \mathbb{RP}^n$ is the tautological real line bundle over $\mathbb{P}(U)$ which is nonorientable.

2 corrected typos

You can get a large class of examples if you look at the real projective space $\mathbb{PR}^n$. \mathbb{RP}^n$. Each point$L\in \mathbb{PR}^n$mathbb{RP}^n$ is a one dimensional subspace in $\mathbb{R}^{n+1}$ and we denote by $P_L$ the orthogonal projection onto $L$. Fix a unit vector $v\in \mathbb{R}^{n+1}$ and denote by $U$ the orthogonal complement in $\mathbb{R}^{n+1}$ of the line $L_v$ spanned by $v$. Consider the function

$$f_v: \mathbb{PR}^n\to\mathbb{R}, mathbb{RP}^n\to\mathbb{R}, \;\; f_v(L)=(P_Lv,v).$$

Note that

$$0 \leq f_v(L)\leq 1,\;\;\forall L.$$

This is a Morse-Bott function with precisely two critical submanifolds: the locus of minima where $f=0$ and consisting of the real projective space $\mathbb{P}(U)\subset \mathbb{PR}^n mathbb{RP}^n$ consisting of lines in $U$, and a unique maximum point, $L_v\in\mathbb{PR}^n$. Note that L_v\in\mathbb{RP}^n$where$f_v(L_v)=1$. The normal bundle of$\mathbb{P}(U)\subset \mathbb{PR}^n mathbb{RP}^n $is the tautological real line bundle over$\mathbb{P}(U)$which is nonorientable. 1 You can get a large class of examples if you look at the real projective space$\mathbb{PR}^n$. Each point$L\in \mathbb{PR}^n$is a one dimensional subspace in$\mathbb{R}^{n+1}$and we denote by$P_L$the orthogonal projection onto$L$. Fix a unit vector$v\in \mathbb{R}^{n+1}$and denote by$U$the orthogonal complement in$\mathbb{R}^{n+1}$of the line$L_v$spanned by$v$. Consider the function $$f_v: \mathbb{PR}^n\to\mathbb{R}, \;\; f_v(L)=(P_Lv,v).$$ Note that $$0 \leq f_v(L)\leq 1,\;\;\forall L.$$ This is a Morse-Bott function with precisely two critical submanifolds: the locus of minima where$f=0$and consisting of the projective space$\mathbb{P}(U)\subset \mathbb{PR}^n $and a unique maximum point,$L_v\in\mathbb{PR}^n$. Note that$f_v(L_v)=1$. The normal bundle of$\mathbb{P}(U)\subset \mathbb{PR}^n $is the tautological line bundle over$\mathbb{P}(U)\$ which is nonorientable.