show/hide this revision's text 3 deleted 11 characters in body

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.
show/hide this revision's text 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.
show/hide this revision's text 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.