2 Calculations for the pull-back connection

In this section, we assume that $\mathbb {K}=\mathbb {R}$ and that $\varphi :B\rightarrow G_k(\mathbb {R}^N)$ is an isometric immersion. Our goal is to describe $R^\nabla $ and $DR^\nabla $ in terms of the classifying map $\varphi $ .

We require some notation. Let $H\subset K\subset G$ equal the triple $O(N-k)\subset O(k)\times O(N-k)\subset O(N)$ . Let $\mathfrak {h}\subset \mathfrak {k}\subset \mathfrak {g}$ denote their Lie algebras. Let $g_0$ be the biinvariant metric on $G=O(N)$ defined as $g_0(X,Y) = \frac 12\text {trace}(X\cdot Y^T)$ for all $A,B\in \mathfrak {g}$ . We will sometime write $\langle A,B\rangle _0$ to mean $g_0(A,B)$ and write $|A|^2_0$ to mean $g_0(A,A)$ . We endow $G_k(\mathbb {R}^N)=G/K$ with the normal homogeneous metric induced by $g_0$ . Denote $\mathfrak {m}=\mathfrak {k}\ominus \mathfrak {h}$ and $\mathfrak {p}=\mathfrak {g}\ominus \mathfrak {k}$ , where “ $\ominus $ ” denotes the $g_0$ -orthogonal complement, so $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}\oplus \mathfrak {p}$ . Let $h:G\rightarrow G/K=G_k(\mathbb {R}^N)$ denote the projection, which is a Riemannian submersion with respect to the above-mentioned metrics.

Let $p\in B$ , and choose $g\in G$ such that $h(g)=\varphi (p)$ . Let $\mathfrak {T}\subset \mathfrak {p}$ denote the subspace such that $h_*(dL_g(\mathfrak {T}))=\varphi _*(T_pB)$ . For any $Z\in T_pB$ , let $\tilde {Z}\in \mathfrak {T}$ denote the unique vector such that $h_*(dL_g(\tilde {Z})) = \varphi _* Z$ . Let $\widetilde {II}:\mathfrak {T}\times \mathfrak {T}\rightarrow \mathfrak {p}\ominus \mathfrak {T}$ denote the lift of the second fundamental form, $II$ , of $\varphi (B)$ . In other words, $h_*(dL_g(\widetilde {II}(\tilde {X},\tilde {Y}))) = II(\varphi _*X,\varphi _*Y)$ for all $X,Y\in T_p B$ .

Notice that the validity of these conditions do not depend on the choice of $g\in h^{-1}(\varphi (p))$ , even though some of the individual terms do.

As we will explain later in this section, $\alpha $ represents the plane spanned by W and V, and Inequalities (1.1) and (2.1) match each other term-for-term in the obvious manner, from which parts (1) and (2) of the proposition follow. Thus, all that is really required to prove Proposition 2.1 is to describe $R^\nabla $ and $DR^\nabla $ in terms of $\varphi $ . The remainder of this section is devoted to this task.

It is already clear that we intend to work in the setting of principal bundles, rather than vector bundles. The total space of the universal principal $O(k)$ -bundle over $G_k(\mathbb {R}^N)$ is the collection of “frames,” i.e., ordered sets of k orthonormal vectors in $\mathbb {R}^N$ :

$$ \begin{align*}F_k(\mathbb{R}^N) = O(N)/O(N-k).\end{align*} $$

Let $\pi :F_k(\mathbb {R}^N)\rightarrow G_k(\mathbb {R}^N)$ denote the projection map, which sends each frame to its span.

This universal principal bundle, $O(k)\hookrightarrow F_k(\mathbb {R}^N)\stackrel {\pi }{\rightarrow }G_k(\mathbb {R}^N)$ , can be re-described as the following homogenous bundle:

(2.2) $$ \begin{align} K/H\hookrightarrow G/H\stackrel{\pi}{\rightarrow}G/K. \end{align} $$

Notice that $\pi $ becomes a Riemannian submersion when $G/H$ and $G/K$ are endowed with the normal homogeneous metrics induced by $g_0$ . Let ${\mathcal {V}}$ and ${\mathcal {H}}$ denote the vertical and horizontal distributions of $\pi $ . We will refer to ${\mathcal {H}}$ as the “universal connection” because:

Consider the chain of Riemannian submersions

$$ \begin{align*}G\stackrel{f}{\rightarrow}G/H=F_k(\mathbb{R}^N)\stackrel{\pi}{\rightarrow}G/K=G_k(\mathbb{R}^N),\end{align*} $$

and denote $h=\pi \circ f$ . Let $\omega _{\mathcal {H}}$ denote the connection form of ${\mathcal {H}}$ , which is an $o(k)$ -valued $1$ -form on $F_k(\mathbb {R}^N)$ . Let $\Omega _{\mathcal {H}}$ denote its curvature form, which is an $o(k)$ -valued $2$ -form on $F_k(\mathbb {R}^N)$ .

Let $g\in G$ be arbitrary, and denote $q=f(g)\in F_k(\mathbb {R}^N)$ . Recall that $L_g:G\rightarrow G$ induces an isometry $F_k(\mathbb {R}^N)\rightarrow F_k(\mathbb {R}^N)$ which preserves ${\mathcal {V}}$ and ${\mathcal {H}}$ . Therefore,

$$ \begin{align*}{\mathcal{H}}_q = \{f_*(dL_gX)\mid X\in\mathfrak{p}\}\,\,\,\text{ and }\,\,\,\ {\mathcal{V}}_q = \{f_*(dL_gV)\mid V\in\mathfrak{m}\}.\end{align*} $$

Notice that for any $V\in \mathfrak {m}\cong o(k)$ , we have

(2.3) $$ \begin{align}\omega_{{\mathcal{H}}}(f_*(dL_gV)) = V,\end{align} $$

simply because the action fields for the principal right $O(k)$ -action on $F_k(\mathbb {R}^N)$ are projections via $f_*$ of left-invariant fields on G.

Proof. We first consider the special case $g=e$ (the identity of G). In this case, let $t\mapsto g(t)$ denote the geodesic in G with $g(0)=e$ and $g'(0)=X$ . Let $t\mapsto Y(t)=dL_{g(t)}Y$ denote the left-invariant extension of Y along this geodesic. Notice that $t\mapsto (f\circ g)(t)$ is a $\pi $ -horizontal geodesic in $F_k(\mathbb {R}^N)$ , and that $t\mapsto \hat {Y}(t) = f_*(Y(t))$ is a $\pi $ -horizontal extension of $f_*Y$ along this horizontal geodesic. Therefore, the O’Neill tensor, A, of $\pi $ at the point q satisfies:

$$ \begin{align*}A(f_*X,f_*Y) = \left(\hat{Y}'(0)\right)^{\mathcal{V}} = \left(f_*(Y'(0))\right)^{\mathcal{V}} = \frac 12 \left(f_*[X,Y]\right)^{\mathcal{V}}= \frac 12 f_*\left([X,Y]^{\mathfrak{m}}\right).\end{align*} $$

Now, we return to the case where $g\in G$ is arbitrary. The previous formula implies:

$$ \begin{align*}A(f_*(dL_gX),f_*(dL_gY)) = \frac 12 f_*\left(dL_g\left([X,Y]^{\mathfrak{m}}\right)\right)\in{\mathcal{V}}_{f(g)}.\end{align*} $$

Together with equation (2.3), this gives:

(2.4) $$ \begin{align}\Omega_{\mathcal{H}}(f_*(dL_g X),f_*(dL_g Y)) &= \omega_{{\mathcal{H}}}(A(f_*(dL_gX),f_*(dL_gY)))\nonumber\\ &=\frac 12 [X,Y]^{\mathfrak{m}}.\nonumber\\[-36pt]\end{align} $$

▪

Let $\pi _P:P\rightarrow B$ be the principal $O(k)$ -bundle obtained as the pull-back via $\varphi $ of the universal frame bundle. Explicitly, $P=\{(p,a)\in B\times F_k(\mathbb {R}^N)\mid \varphi (p)=\pi (a)\}$ with projection $\pi _P(p,a) = p$ . Let $\overline {\varphi }:P\rightarrow F_k(\mathbb {R}^N)$ denote the corresponding bundle homomorphism, defined as $\overline {\varphi }(p,a)=a$ .

Let $\omega $ denote the connection form of the principal connection in P obtained as the pull-back via $\varphi $ of the universal connection in the universal frame bundle. Let $\Omega $ denote the curvature form of $\omega $ . Thus, $\omega $ is an $o(k)$ -valued 1-form on P, and $\Omega $ is an $o(k)$ valued $2$ -form on P.

Let $\pi _E\kern-.5pt :\kern-.5pt E\rightarrow\kern-.5pt B$ denote the vector bundle associated to P. Explicitly, $E=P\kern-.5pt \times _{O(k)}\mathbb {R}^k$ . Let $\nabla $ the connection in E associated to $\omega $ , and let $R^\nabla $ denote its curvature tensor. Let $p\in B$ . We introduce the following notation.

Let $X,Y\in T_p B$ be orthonormal, and let $W,V\in E_p$ be orthonormal. For simplicity, we initially assume that $\varphi (p) = h(e)\in G_k(\mathbb {R}^N)$ . Let $a=f(e)\in F_k(\mathbb {R}^N)$ , and let $\tilde {a}=(a,p)\in P$ . Notice that $\mathbb {R}^k$ can be identified with the fiber $E_p$ via the map which sends $\hat {U}\in \mathbb {R}^k$ to $U=\tilde {a}\diamond \hat {U}\in E_p$ . Let $\hat {W},\hat {V}\in \mathbb {R}^k$ be associated with $W,V\in E_p$ in this way. There exists a unique $\alpha \in o(k)$ such that $\alpha \cdot \hat {W}=\hat {V}$ , $\alpha \cdot \hat {V}=-\hat {W}$ , and $\alpha \cdot \hat {U}=0$ for all $\hat {U}\in \mathbb {R}^k$ orthogonal to $\text {span}\{\hat {W},\hat {V}\}$ , where the dots denote matrix multiplication. We claim that

(2.5) $$ \begin{align}\langle \beta\cdot\hat{W},\hat{V}\rangle = \langle\beta,\alpha\rangle_0 \,\,\,\,\,\,\,\,\,\text{ for all }\beta\in o(k).\end{align} $$

To see this, just choose an ordered orthonormal basis of $\mathbb {R}^k$ beginning with $\hat {W},\hat {V}$ , and then express $\alpha ,\beta $ in terms of the corresponding standard basis for $o(k)$ .

For any $Z\in T_p B$ , we let $\overline {Z}\in T_{\tilde {a}}P$ denote its $\pi _P$ -horizontal lift, and we let $\tilde {Z}\in \mathfrak {p}$ denote the unique vector such that $h_*\tilde {Z} = \varphi _* Z$ . In the following calculation, the maximum is taken over all unit-length $Z\in T_pB$ :

$$ \begin{align*} |R^\nabla(W,V)X| & = \max\langle R^\nabla(W,V)X,Z\rangle = \max\langle R^\nabla(X,Z)W,V\rangle\\ & = \max\langle \Omega(\overline{X},\overline{Z})\cdot \hat{W},\hat{V}\rangle = \text{max}\langle\Omega(\overline{X},\overline{Z}),\alpha\rangle_0 \\ & = \text{max}\langle\Omega_{\mathcal{H}} (\overline{\varphi}_*\overline{X},\overline{\varphi}_*\overline{Z}),\alpha\rangle_0 = \frac 12\text{max}\langle[\tilde{X},\tilde{Z}]^{\mathfrak{m}},\alpha\rangle_0 \\ & = \frac 12\text{max}\langle[\tilde{X},\tilde{Z}],\alpha\rangle_0 = \frac 12\text{max}\langle[\tilde{X},\alpha],\tilde{Z}\rangle_0= \frac 12|[\tilde{X},\alpha]^{\mathfrak{T}}|_0. \end{align*} $$

The last equality uses that $\varphi $ is an isometric immersion, so maximizing over all unit-length $Z\in T_p B$ is the same as maximizing over all unit-length $\tilde {Z}\in \mathfrak {T}$ . Recall that where $\mathfrak {T}\subset \mathfrak {p}$ is the subspace such that $h_*(\mathfrak {T})=\varphi _*(T_pB)$ . In summary,

(2.6) $$ \begin{align} |R^\nabla(W,V)X| = \frac{1}{2}{|[\tilde{X},\alpha]^{\mathfrak{T}}|}_0. \end{align} $$

It remains to express the expression $\langle (D_ZR^\nabla )(X,Y)W,V\rangle $ in terms of the geometry of $\varphi $ . For this, let $t\mapsto c(t)$ denote the geodesic in B with $c(0)=p$ and $c'(0)=Z$ . Let $t\mapsto \overline {c}(t)$ denote its $\pi _P$ -horizontal lift beginning at $ \tilde {a} \in P$ . We can write $\overline {c}(t) = (c(t),\beta (t))$ where $t\mapsto \beta (t)$ is the $\pi $ -horizontal lift of $t\mapsto \varphi (c(t))$ to $F_k(\mathbb {R}^N)$ beginning at $\beta (0)=a$ . Let $t\mapsto g(t)$ be the h-horizontal lift to G of $t\mapsto \varphi (c(t))$ beginning at $g(0)=e$ . Define $W(t)=\overline {c}(t)\diamond \hat {W}\in E_{c(t)}$ and $V(t)=\overline {c}(t)\diamond \hat {V}\in E_{c(t)}$ . Notice that $V(t)$ and $W(t)$ are parallel because $\overline {c}(t)$ is horizontal.

Let $X(t)$ and $Y(t)$ denote the parallel extensions of $X,Y$ along $t\mapsto c(t)$ . Let $\overline {X}(t)$ and $\overline {Y}(t)$ denote the $\pi _P$ -horizontal lifts of these fields along $t\mapsto \overline {c}(t)$ . Let $\tilde {X}(t)$ and $\tilde {Y}(t)$ denote the h-horizontal lifts along $t\mapsto g(t)$ of the fields $t\mapsto \varphi _*X(t)$ and $t\mapsto \varphi _*Y(t)$ . Notice that

$$ \begin{align*}R^\nabla(X(t),Y(t))W(t) = \overline{c}(t)\diamond\left(\Omega(\overline{X}(t),\overline{Y}(t))\cdot\hat{W}\right) \in E_{c(t)}.\end{align*} $$

In the following calculation, we will use $\frac {D}{dt}$ to denote covariant differentiation with respect to $\nabla $ , and $\frac {d}{dt}$ to denote the usual differentiation of a path of vectors in the Euclidean spaces $\mathbb {R}^k$ and $o(k)$ , and prime ( $'$ ) to denote covariant differentiation with respect to the Levi Civita connection in $(G,g_0)$ . With this notation, we have

$$ \begin{align*} \langle (D_Z R^\nabla)(X,Y)W,V\rangle & = \left\langle \frac{D}{dt}\Big|_{t=0} \overline{c}(t)\diamond\left(\Omega(\overline{X}(t),\overline{Y}(t)) \cdot\hat{W}\right),V\right\rangle \\& = \left\langle \overline{c}(0)\diamond\left(\frac{d}{dt}\Big|_{t=0} \Omega(\overline{X}(t),\overline{Y}(t))\cdot\hat{W}\right),V\right\rangle \\& = \left\langle \left(\frac{d}{dt}\Big|_{t=0} \Omega(\overline{X}(t),\overline{Y}(t))\right)\cdot\hat{W},\hat{V}\right\rangle \\& = \left\langle \frac{d}{dt}\Big|_{t=0} \Omega(\overline{X}(t),\overline{Y}(t)),\alpha\right\rangle_0\\& = \left\langle \frac{d}{dt}\Big|_{t=0} \Omega_{{\mathcal{H}}}(\overline{\varphi}_*\overline{X}(t), \overline{\varphi}_*\overline{Y}(t)),\alpha\right\rangle_0 \\& = \frac 12 \left\langle \frac{d}{dt}\Big|_{t=0} \left[dL_{g(t)}^{-1}\tilde{X}(t) ,dL_{g(t)}^{-1}\tilde{Y}(t)\right]^{\mathfrak{m}},\alpha\right\rangle_0 \\& = \frac 12 \bigg\langle \left[\frac{d}{dt}\Big|_{t=0} dL_{g(t)}^{-1}\tilde{X}(t),\tilde{Y}\right]^{\mathfrak{m}}\\& \quad + \left[ \tilde{X},\frac{d}{dt}\Big|_{t=0}dL_{g(t)}^{-1}\tilde{Y}(t)\right]^{\mathfrak{m}},\alpha\bigg\rangle_0. \end{align*} $$

To interpret these terms, let $\{E_i\}$ denote an orthonormal basis of $\mathfrak {p}$ , and let $\{E_i(t)\}$ denote their left-invariant extensions along $g(t)$ ; that is, $E_i(t)=dL_{g(t)}E_i$ . Then,

$$ \begin{align*} \frac{d}{dt}\Big|_{t=0}dL_{g(t)}^{-1}\tilde{Y}(t) & = \frac{d}{dt}\Big|_{t=0}\sum\langle dL_{g(t)}^{-1}\tilde{Y}(t),E_i\rangle_0 E_i = \frac{d}{dt}\Big|_{t=0}\sum\langle\tilde{Y}(t),E_i(t)\rangle_0 E_i \\[4pt] & = \sum\langle\tilde{Y}'(0),E_i\rangle_0 E_i + \sum\langle\tilde{Y},E_i'(0)\rangle_0 E_i\\ & = \tilde{Y}'(0) + \frac 12 \sum\langle\tilde{Y},[\tilde{Z},E_i]\rangle_0 E_i \\ & = \tilde{Y}'(0) - \frac 12 \sum\langle E_i,[\tilde{Z},\tilde{Y}]\rangle_0 E_i\\ & = \tilde{Y}'(0) - \frac 12[\tilde{Z},\tilde{Y}]. \end{align*} $$

Since $t\mapsto Y(t)$ is a parallel vector field along the geodesic $t\mapsto c(t)$ in B, we have $\tilde {Y}'(0) = \widetilde {II}(\tilde {Z},\tilde {Y}) + A_h(\tilde {Z},\tilde {Y})$ , where $A_h$ denotes the O’Neill tensor of h. In summary,

$$ \begin{align*} \left[\tilde{X},\frac{d}{dt}\Big|_{t=0}dL_{g(t)}^{-1}\tilde{Y}(t)\right]^{\mathfrak{m}} & = \left[\tilde{X},\tilde{Y}'(0)-\frac 12[\tilde{Z},\tilde{Y}]\right]^{\mathfrak{m}}\\ & = \left[\tilde{X},\widetilde{II}(\tilde{Z},\tilde{Y})+A_h(\tilde{Z},\tilde{Y}) - \frac 12[\tilde{Z},\tilde{Y}]\right]^{\mathfrak{m}}\\ & = \left[\tilde{X},\widetilde{II}(\tilde{Z},\tilde{Y})\right]^{\mathfrak{m}}. \end{align*} $$

The last equality follows from the fact that $\tilde {X}\in \mathfrak {p}$ while $A_h(\tilde {Z},\tilde {Y})\in \mathfrak {k}$ and $[\tilde {Z},\tilde {Y}]\in \mathfrak {k}$ (because $\mathfrak {k}\subset \mathfrak {g}$ is a symmetric pair). We similarly have that

$$ \begin{align*}\left[\tilde{Y},\frac{d}{dt}\Big|_{t=0}dL_{g(t)}^{-1}\tilde{X}(t)\right]^{\mathfrak{m}} = \left[\tilde{Y},\widetilde{II}(\tilde{Z},\tilde{X})\right]^{\mathfrak{m}}.\end{align*} $$

Therefore, since $\alpha \in \mathfrak {m}$ , we have

(2.7) $$ \begin{align}\langle (D_Z R^\nabla)(X,Y)W,V\rangle = \frac 12\left\langle \left[\tilde{X},\widetilde{II}(\tilde{Z},\tilde{Y})\right] -\left[\tilde{Y},\widetilde{II}(\tilde{Z},\tilde{X})\right], \alpha\right\rangle_0.\end{align} $$

3 Totally geodesic classifying maps

In this section, we assume that $\mathbb {K}=\mathbb {R}$ and that the classifying map $\varphi :B\rightarrow G_k(\mathbb {R}^N)$ is a totally geodesic isometric imbedding. By Proposition 2.1, this implies that $\nabla $ is parallel. If, additionally, B has positive sectional curvature, then $\nabla $ induces a connection metric of nonnegative curvature on E [Reference Strake and Walschap15].

Since $G_k(\mathbb {R}^N)$ contains many totally geodesic submanifolds (which have not yet been fully classified) including many with positive curvature, this might appear to be a hopeful source for topologically new examples of vector bundles which admit metrics of nonnegative curvature. But we will explain in this section why no new examples can be obtained in this way.

First notice that B is a symmetric space because it is a totally geodesic submanifold of the symmetric space $G_k(\mathbb {R}^N)=G/K$ (here, as before, $H\subset K\subset G$ denotes the triple $O(N-k)\subset O(k)\times O(N-k)\subset O(N)$ ). More precisely, $B=G'/K'$ where $G'\subset G$ and $K'=G'\cap K$ . Let $\rho ':K'\rightarrow O(k)$ denote the composition of the inclusion map into K with the projection onto the first factor. The following was observed by Rigas in [Reference Rigas9].

Proof. Let $\rho :K\rightarrow O(k)$ denote the projection onto the first factor. The universal principal bundle $O(k)\hookrightarrow F_k(\mathbb {R}^N)\rightarrow G_k(\mathbb {R}^N)$ was re-described in equation (2.2) as the homogenous bundle $K/H\hookrightarrow G/H\rightarrow G/K.$ This homogeneous bundle can again be re-described as the associated bundle $K/H\hookrightarrow G\times _{\rho }(K/H)\rightarrow G/K.$ This implies that the universal vector bundle over $G_k(\mathbb {R}^N)$ can be re-described as $\mathbb {R}^k\hookrightarrow G\times _\rho \mathbb {R}^k\rightarrow G/K$ . Consider the following commutative diagram, in which the right arrows denote the natural inclusion maps:

the bundle on the right is the universal vector bundle over $G_k(\mathbb {R}^N)=G/K$ , so the bundle on the left is isomorphic to the pull-back bundle $\pi _E:E\rightarrow B$ , as desired.▪

In conclusion, if the classifying map $\varphi :B\rightarrow G_k(\mathbb {R}^N)$ is a totally geodesic isometric imbedding, then $\nabla $ is parallel, but in this case the bundle is isomorphic to an associated bundle, which trivially admits a submersion metric of nonnegative sectional curvature.

4 The cases of ${\mathbb {C}}^1$ and ${\mathbb {H}}^1$ bundles

The primary technical difficulty in applying Proposition 2.1 is that “bracketing with $\alpha $ ” is difficult to interpret geometrically in general. However, in this section, we will provide a very natural geometric interpretation in the special case where $k=1$ and $\mathbb {K}\in \{{\mathbb {C}},{\mathbb {H}}\}$ (so that $\varphi :B\rightarrow {\mathbb {CP}}^{N-1}$ or $\varphi :B\rightarrow {\mathbb {HP}}^{N-1}$ ). Theorems 1.1, 1.5, and 1.6 will follow from this interpretation.

Even though we assumed in the Section 2 that $\mathbb {K}=\mathbb {R}$ , almost all of the calculations generalize in the obvious way to the cases $\mathbb {K}\in \{{\mathbb {C}},{\mathbb {H}}\}$ . The only exception involves the manner in which $\alpha $ was chosen to represent a particular plane in the fiber. Recall that we were given arbitrary orthonormal vectors $W,V\in \mathbb {R}^k$ , and we were able to choose $\alpha \in o(k)$ to represent the plane $\text {span}\{W,V\}$ in the sense that

$$ \begin{align*}\langle \beta\cdot W,V\rangle = \langle\beta,\alpha\rangle_0 \,\,\,\,\,\,\,\,\,\text{ for all }\beta\in o(k).\end{align*} $$

To generalize this proof to the case $\mathbb {K}={\mathbb {C}}$ (respectively $\mathbb {K}={\mathbb {H}}$ ), we would be given arbitrary $\mathbb {R}$ -orthonormal vectors $W,V\in \mathbb {K}^k$ , and we would need to choose $\alpha \in u(k)$ (respectively $\alpha \in sp(k)$ ) so $\langle \beta \cdot W,V\rangle _{\mathbb {R}} = \langle \beta ,\alpha \rangle _0$ for all $\beta \in u(k)$ (respectively $\beta \in sp(k)$ ). Unfortunately, this is not generally possible unless we additionally assume that $W\perp \text {span}\{\mathfrak {J}V\mid \mathfrak {J}\in \mathcal {J}\}$ .

However, when $k=1$ , there is no trouble with choosing $\alpha $ as desired. In fact, the choice $\alpha =V\cdot \overline {W}$ works, and all of the calculations in the previous section go through.

More specifically, when $k=1$ and $\mathbb {K}={\mathbb {C}}$ (so that $\varphi :B\rightarrow {\mathbb {CP}}^{N-1}$ ), the chain $H\subset K\subset G$ from Section 2 becomes $SU(N-1)\subset S(U(1)\times U(N-1))\subset SU(N)$ , and $\mathfrak {m}=u(1)$ is spanned by a unique (up to sign) unit-length vector $\alpha \in \mathfrak {m}$ . It is not hard to see that for any $q\in G/K={\mathbb {CP}}^{N-1}$ , the map $\text {ad}_\alpha :\mathfrak {p}\rightarrow \mathfrak {p}$ (which sends $\tilde {X}$ to $[\alpha ,\tilde {X}]$ ) induces an involution of $T_q({\mathbb {CP}}^{N-1})$ that is well-defined in the sense that it is independent of the choice of $g\in h^{-1}(q)$ through which the lift $\tilde {X}$ is defined as in Proposition 2.1. In fact, this is one natural way in which to define the standard almost complex structure on ${\mathbb {CP}}^{N-1}$ . Therefore, in Proposition 2.1, bracketing with $\alpha $ can be interpreted as applying the almost-complex structure J.

When $k=1$ and $\mathbb {K}={\mathbb {H}}$ (so that $\varphi :B\rightarrow {\mathbb {HP}}^{N-1}$ ), the chain $H\subset K\subset G$ from the previous section becomes $Sp(N-1)\subset Sp(1)\times Sp(N-1)\subset Sp(N)$ . Identify $\mathcal {J}=\{I,J,K\}$ with an oriented orthonormal basis of $\mathfrak {m}=sp(1)=\text {Im}({\mathbb {H}})$ . For any $q\in G/K={\mathbb {HP}}^{N-1}$ , the triple of maps $\text {ad}_I,\text {ad}_J,\text {ad}_K:\mathfrak {p}\rightarrow \mathfrak {p}$ induces a triple of involutions of $T_q({\mathbb {HP}}^{N-1})$ which satisfy the familiar properties of an almost quaternionic structure. Changing to a different $g\in h^{-1}(q)$ has the effect of conjugating to a different oriented orthonormal basis of $\mathfrak {m}$ , so the family of triples:

$$ \begin{align*}\{\{\text{ad}_I,\text{ad}_J,\text{ad}_K\}\mid\{I,J,K\} \text{ is an oriented orthonormal basis of }\mathfrak{m} \}\end{align*} $$

determines a well-defined family of triples of involutions of $T_q{\mathbb {HP}}^{N-1}$ . This is one way to define the natural almost quaternionic structure on ${\mathbb {HP}}^{N-1}$ . Recall that on an almost quaternionic manifold, a choice of basis $\{I,J,K\}$ for the almost quaternionic structure generally only exists locally, which is reflected in the dependence on $g\in h^{-1}(q)$ described above. In any case, bracketing with all possible $\alpha \in \mathfrak {m}$ can be interpreted in Proposition 2.1 as applying all possible elements of $\text {span}\{\mathcal {J}\}$ .

Proof of Theorem 1.1

Recall that $\nabla $ is fat if and only if $|R^\nabla (W,V)X|>0$ for all $p\in B$ , nonzero $X\in T_pB$ and all orthonormal $W,V\in E_p$ . If $\varphi $ is not an immersion, then there exits $p\in B$ and $X\in T_pB$ such that $\varphi _*X=0$ , which implies that $R^\nabla (W,V)X=0$ for any choice of $W,V$ . Thus, $\nabla $ is not fat.

Next assume that $\varphi $ is an immersion, and choose the pull-back metric for B. Let $\alpha \in \mathfrak {m}$ represent the plane $\text {span}\{W,V\}$ in the sense of equation (2.5). Let $\mathfrak {J}\in \text {span}\{\mathcal {J}\}$ represent $\alpha $ as described previously in this section. By equation (2.6),

$$ \begin{align*}2\left|R^\nabla(W,V)X\right| = \left|[\tilde{X},\alpha]^{\mathfrak{T}}\right|{}_0 = \left|(\mathfrak{J}X)^{\varphi_*(T_pB)}\right|\geq|X|\cdot\cos(\theta(X)),\end{align*} $$

so $\nabla $ is fat if and only if $\theta (X)<\pi /2$ for all nonzero $X\in TM$ .▪

Proof of Theorem 1.5

The connection $\nabla $ is parallel if and only if the following equals zero for all $p\in B$ , $X,Y,Z\in T_pB$ and $W,V\in E_p$ :

$$ \begin{align*}2 \langle (D_Z R^\nabla)(X,Y)W,V\rangle & = \left\langle \left[\tilde{X},\widetilde{II}(\tilde{Z},\tilde{Y})\right] -\left[\tilde{Y},\widetilde{II}(\tilde{Z},\tilde{X})\right], \alpha\right\rangle_0\\ & = \left\langle \left[\alpha,\tilde{X}\right],\widetilde{II}(\tilde{Z},\tilde{Y})\right\rangle_0 -\left\langle \left[\alpha,\tilde{Y}\right],\widetilde{II}(\tilde{Z},\tilde{X})\right\rangle_0\\ & = \left\langle \mathfrak{J}X,II(Z,Y)\right\rangle -\left\langle \mathfrak{J}Y,II(Z,X)\right\rangle\\ & = \left\langle S_{(\mathfrak{J}X)^\perp}Y,Z\right\rangle - \left\langle S_{(\mathfrak{J}Y)^\perp}X,Z\right\rangle\\ & = \left\langle S_{(\mathfrak{J}X)^\perp}Y - S_{(\mathfrak{J}Y)^\perp}X,Z\right\rangle, \end{align*} $$

where $\alpha \in \mathfrak {m}$ represent the plane $\text {span}\{W,V\}$ in the sense of equation (2.5), and $\mathfrak {J}\in \text {span}\{\mathcal {J}\}$ represents $\alpha $ as described previously in this section. Furthermore, $\nabla $ is radially symmetric if and only if the above is true in the special case $Z=X$ .▪

Theorem 1.6 follows immediately from the calculations of the previous two proofs. It remains only to prove Corollary 1.7.

Proof of Corollary 1.7

The inequality of Theorem 1.6 is

$$ \begin{align*}\langle S_{(\mathfrak{J}X)^\perp}Y - S_{(\mathfrak{J}Y)^\perp}X,X\rangle^2\leq k_B(X,Y)\cdot|\text{proj}(\mathfrak{J}X)|^2.\end{align*} $$

The terms of this inequality at a point $p\in B$ are bounded as follows:

(4.1) $$ \begin{align} \langle S_{(\mathfrak{J}X)^\perp}Y - S_{(\mathfrak{J}Y)^\perp}X,X\rangle^2 & \leq 4|S(p)|^2\sin^2\theta(p).\notag \\ k_B(X,Y) & = k_{\mathbb{KP}^N}(X,Y) +\langle II(X,X),II(Y,Y)\rangle - |II(X,Y)|^2\notag \\ & \geq k_{\mathbb{KP}^N}(X,Y) - 2|S(p)|^2\notag \\ & \geq 1/4 - 2|S(p)|^2.\\ |\text{proj}(\mathfrak{J}X)|^2 & \geq \cos^2\theta(p).\notag \end{align} $$

Thus, the inequality is satisfied if

$$ \begin{align*}4|S(p)|^2\sin^2\theta(p) \leq (1/4 - 2|S(p)|^2)\cos^2\theta(p),\end{align*} $$

which can be re-expressed as

$$ \begin{align*}|S(p)|^2\leq\frac{1}{16\tan^2\theta(p) + 8}.\\[-40pt] \end{align*} $$

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5 Well studied classes of immersions into ${\mathbb {CP}}^N$ and ${\mathbb {HP}}^N$

Theorems 1.1, 1.5, and 1.6 empower one to construct connections with certain natural geometric properties in $\mathbb {K}^1$ -vector bundles over B (with $\mathbb {K}\in \{{\mathbb {C}},{\mathbb {H}}\}$ ) by constructing immersions of B into $\mathbb {KP}^{N-1}$ that satisfy certain hypotheses. There is a large body of literature on immersed submanifolds of projective spaces with natural properties. Some of these properties overlap the hypotheses required by our theorems. In this section, we review some of the literature and discuss its relevance to the search for nice connections in $\mathbb {K}^1$ bundles.

As mentioned previously, if $\varphi $ is an isometric complex/quaternionic immersion, then $\nabla $ is fat and parallel. If additionally B has positive curvature, then Inequality (1.2) in Theorem 1.6 is strictly satisfied, so $\nabla $ induces a connection metric of nonnegative curvature in E and of positive curvature in $E^1$ . Since these are strong conclusions, it is worthwhile to begin by surveying some of the relevant literature on isometric complex/quaternionic immersions. We start with the case $\mathbb {K}={\mathbb {C}}$ .

Calabi’s rigidity theorem from [Reference Calabi1] says that if $f_1,f_2:B\rightarrow {\mathbb {CP}}^N$ are both isometric complex immersions, then there exists a unitary transformation U of ${\mathbb {CP}}^N$ such that $f_2=U\circ f_1$ . Calabi further classified all isometric complex imbeddings of ${\mathbb {CP}}^n(c_1)$ into ${\mathbb {CP}}^N(c_2)$ , where $c_1,c_2$ denote the constant holomorphic curvature. For any fixed n, he proved there exists a countably infinite family of imbeddings, $f_i:{\mathbb {CP}}^n(c/i)\rightarrow {\mathbb {CP}}^{N_i}(c)$ , where $i\in \mathbb {Z}^+$ and $N_i=\frac {(n+i)!}{n!i!}-1$ . The map $f_i$ is sometimes called the $i{\text {th}}$ Veronese imbedding. It is not totally geodesic if $i>1$ . Each $f_i$ induces a parallel fat connection in the pulled-back ${\mathbb {C}}^1$ -bundle over ${\mathbb {CP}}^n$ , and the total space, $E_i^1$ , of the corresponding circle-bundle inherits a connection metric of positive sectional curvature.

These examples are not new. The main result of [Reference Guijarro, Sadun and Walschap5] implies that any vector bundle over the symmetric space ${\mathbb {CP}}^n=SU(n+1)/S(U(1)\times U(n))$ with a parallel connection must be isomorphic to an associated bundle, which means it has the form $SU(n+1)\times _{\rho }{\mathbb {C}}\rightarrow {\mathbb {CP}}^n$ for some representation $\rho :S(U(1)\times U(n))\rightarrow U(1)$ . There is a one parameter family of such representations coming from powers of the determinant of $A\in U(n)\cong S(U(1)\times U(n))$ : $\rho _j(A)=\det (A)^j$ . The total space of each circle bundle therefore has the following form for some $j\in \mathbb {Z}^+$ :

(5.1) $$ \begin{align}E^1_j=SU(n+1)\times_{\rho_j}U(1),\end{align} $$

which can be shown to be diffeomorphic to the lens space $S^{2n+1}/\mathbb {Z}_j$ .

Nakagawa and Takagi in [Reference Nakagawa and Takagi7] classified the isometric complex imbeddings of all other compact simply connected irreducible Hermitian symmetric space into ${\mathbb {CP}}^N$ . For each such space, they obtained a countably infinite families of imbeddings analogous to the Veronese imbeddings. Notice that ${\mathbb {CP}}^n$ is the only such space with positive curvature, so no new examples of connection metrics with positive curvature in circle bundles could be obtained by pulling back the universal connection via these imbeddings. As above, these pulled-back connections are parallel and fat, and the bundles are associated bundles.

More recently, Di Scala, Ishi, and Loi studied isometric complex immersions of the form $f:B\rightarrow {\mathbb {CP}}^N(1)$ , where B is a homogeneous Kähler manifold [Reference Di Scala, Ishi and Loi4]. They proved that f must be an imbedding and that B must be simply connected. Moreover, they conjectured that (some rescaling of) any simply connected homogeneous Kähler manifold, B, whose associated Kähler form is integral must admit an isometric complex imbedding into ${\mathbb {CP}}^N(1)$ for some N. This conjecture would imply that over each such space there exists a ${\mathbb {C}}^1$ -bundle that admits a parallel fat connection.

There is no classification of the isometric complex immersions $f:B\rightarrow {\mathbb {CP}}^N(1)$ for which B has positive or nonnegative sectional curvature, except under added hypotheses. For example, If B complex dimension $\geq 2$ and sectional curvature $>1/8$ , then Ros and Verstraelen proved that $f(B)$ must be totally geodesic [Reference Ros and Verstraelen11]. If B has positive sectional curvature and has holomorphic curvature $\geq 1/2$ , then Ros proved that f must be a one of a list of standard imbeddings [Reference Ros10]. If B has nonnegative sectional curvature and has complex codimension less than its complex dimension, then Shen obtained a similar conclusion [Reference Shen14]. But without any added hypotheses, no classification is known. Any new example would be interesting, especially if B had positive sectional curvature, for then the pulled back circle bundle over B would inherit positive sectional curvature as well.

There are other conditions on immersions (more general than the complex/quaternionic condition) that imply bounded Wirtinger angles. For example, an immersion $\varphi :B\rightarrow \mathbb {KP}^N$ is called called slant if $\theta $ is constant, or semi-slant if its tangent bundle decomposes into two sub-bundles on which $\theta $ is constant, respectively at zero and at another value. Slant submanifolds were defined by Chen, who summarized the early results in his book [Reference Chen2]. The pull-back of any proper slant (or semi-slant) immersion would yield a $\mathbb {K}^1$ bundle with a fat connection. The literature on slant and semi-slant submanifolds consists primarily of rigidity results. However, Maeda et al. in [Reference Maeda, Ohnita and Udagawa6] constructed examples of slant submanifolds of ${\mathbb {CP}}^N$ , including families of proper slant imbeddings of ${\mathbb {CP}}^n$ into ${\mathbb {CP}}^N$ which generalize the Veronese imbeddings.