1 Introduction and main results

Let (M, g) be a compact Riemannian n-manifold and let \({{\,\mathrm{sec}\,}}_g\) be the sectional curvature of the metric. We often abuse notation and denote the Riemannian metric by (M, g) as well. For each 2-plane

$$\begin{aligned} \sigma \in {{\,\mathrm{Gr}\,}}_2(T_pM) = \{X\wedge Y\in {\Lambda }^2 T_pM : ||X\wedge Y||^2 = 1\},\end{aligned}$$
(1.1)

let \(\sigma ^\perp \subset T_pM\) be its orthogonal complement. That is, there is a g-orthogonal direct sum decomposition \(\sigma \oplus \sigma ^{\perp } = T_pM\) at a point \(p\in M\).

Definition 1

The biorthogonal curvature of a 2-plane \(\sigma \in {{\,\mathrm{Gr}\,}}_2(T_pM)\) is

$$\begin{aligned} {{\,\mathrm{sec}\,}}_g^{\perp }(\sigma ):= \underset{\begin{array}{c} \sigma '\in {{\,\mathrm{Gr}\,}}_2(T_pM)\\ \sigma ' \subset \sigma ^{\perp } \end{array}}{{{\,\mathrm{min}\,}}}\frac{1}{2}({{\,\mathrm{sec}\,}}_g(\sigma ) + {{\,\mathrm{sec}\,}}_g(\sigma ')) \end{aligned}$$
(1.2)

(cf. [3, Section 5.4]). We say that (M, g) has positive biorthogonal curvature \({{\,\mathrm{sec}\,}}_g^\perp > 0\) if (1.2) is positive for every \(\sigma \in {{\,\mathrm{Gr}\,}}_2(T_pM)\) at every point \(p\in M\).

A stronger curvature condition is the following. Choose a distance on the Grassmanian bundle \({{\,\mathrm{Gr}\,}}_2(TM)\) that induces the standard topology.

Definition 2

The distance curvature of a 2-plane \(\sigma \subset T_pM\) is

$$\begin{aligned} {{\,\mathrm{sec}\,}}_g^{\theta }(\sigma ):= \underset{\begin{array}{c} \sigma '\in {{\,\mathrm{Gr}\,}}_2(T_pM)\\ {{\,\mathrm{dis}\,}}(\sigma , \sigma ')\ge \theta \end{array}}{{{\,\mathrm{min}\,}}}\frac{1}{2}({{\,\mathrm{sec}\,}}_g(\sigma ) + {{\,\mathrm{sec}\,}}_g(\sigma '))\end{aligned}$$
(1.3)

for each \(\theta > 0\) (cf. [3, Section 5.2]). We say that \((M, g^\theta )\) has positive distance curvature \({{\,\mathrm{sec}\,}}_{g^\theta } > 0\) if for every \(\theta > 0\), there is a Riemannian metric \((M, g^\theta )\) for which (1.3) is positive for every \(\sigma \in {{\,\mathrm{Gr}\,}}_2(T_pM)\) at every point \(p\in M\).

Bettiol [4] classified up to homeomorphism closed simply connected 4-manifolds that admit a Riemannian metric of positive biorthogonal curvature by constructing metrics of positive distance curvature on \(S^2\times S^2\) [2, Theorem, Proposition 5.1], [3, Theorem 6.1], and showing that positive biorthogonal curvature is a property that is closed under connected sums [3, Proposition 7.11], [4, Proposition 3.1].

In this paper, we extend Bettiol’s results to dimension five. More precisely, we build upon Bettiol’s work and show that an application of a first-order conformal deformation to Wilking’s metric \((S^2\times S^3, g_W)\) of almost-positive sectional curvature [11] yields the main result of this note.

Theorem A

For every \(\theta > 0\), there is a Riemannian metric \((S^2\times S^3, g^{\theta })\) such that

  1. (a)

    \({{\,\mathrm{sec}\,}}_{g^\theta }^\theta > 0;\)

  2. (b)

    there is a limit metric \(g^0\) such that \(g^{\theta }\rightarrow g^0\) in the \(C^k\)-topology as \(\theta \rightarrow 0\) for \(k\ge 0;\)

  3. (c)

    \(g^\theta \) is arbitrarily close to Wilking’s metric \(g_W\) of almost-positive curvature in the \(C^k\)-topology for \(k\ge 0;\)

  4. (d)

    \({{\,\mathrm{Ric}\,}}_{g^{\theta }} > 0;\)

  5. (e)

    there is a 2-plane \(\sigma \in {{\,\mathrm{Gr}\,}}_2(T_pS^2\times S^3)\) with \({{\,\mathrm{sec}\,}}_{g^{\theta }}(\sigma ) < 0;\)

In particular, there is a Riemannian metric of positive biorthogonal curvature on \(S^2\times S^3\).

The next corollary is a consequence of coupling Theorem A with a classification result of Smale [8].

Corollary B

Every closed simply connected 5-manifold with torsion-free homology and zero second Stiefel–Whitney class admits a Riemannian metric of positive biorthogonal curvature.

The hypothesis imposed on the homology and the second Stiefel–Whitney class of the manifolds of Corollary B are technical in nature; cf. Remark 2. Indeed, an examination of the canonical metric on the Wu manifold yields the following proposition.

Proposition C

The symmetric space metric \((\mathrm {SU}(3)/\mathrm {SO}(3), g)\) has positive biorthogonal curvature.

The Wu manifold has second homology group of order two and nontrivial second Stiefel–Whitney class.

2 Constructions of Riemannian metrics of positive biorthogonal curvature

2.1 Wilking’s metric of almost-positive curvature on \(S^2\times S^3\)

We follow the exposition in [11, Section 5] to describe Wilking’s construction of a metric of almost-positive curvature on the product of projective spaces \(\mathbb {R}P^2\times \mathbb {R}P^3\) and its pullback to \(S^2\times S^3\) under the covering map; see [12, Section 5] for a discussion relating these two constructions.

The unit tangent sphere bundle of the 3-sphere

$$\begin{aligned} T_1(S^3)=S^2\times S^3, \end{aligned}$$
(2.1)

embeds into \(\mathbb {R}^4\times \mathbb {R}^4=\mathbb {H}\times \mathbb {H}\) as a pair of orthogonal unit quaternions

$$\begin{aligned} S^3\times S^2 = \{(p,v)\in \mathbb {H}\times \mathbb {H}:|p|=|v|=1, \langle p,v \rangle = 0\}\subset \mathbb {H}\times \mathbb {H}, \end{aligned}$$
(2.2)

where \(\langle x,y \rangle = \mathrm {Re}(\bar{x}y)\), \(|x|^2=\langle x,x \rangle \), and \(\bar{x}\) denotes the quarternion conjugation of x. The group \(G=\mathrm {Sp(1)}\times \mathrm {Sp(1)} \simeq S^3\times S^3\) acts on \(S^2\times S^3\) by

$$\begin{aligned} (q_1,q_2)\star (p,v)=(q_1 p \bar{q}_2,q_1 v \bar{q}_2) \end{aligned}$$
(2.3)

for \(q_1, q_2 \in \mathrm {Sp(1)}\) and \((p,v) \in S^2 \times S^3\). This action is effectively free and transitive. The isotropy group of the point \((1,i)\in S^2\times S^3\) is

$$\begin{aligned} H=\{(e^{i\phi },e^{i\phi })\in \mathrm {Sp(1)}\times \mathrm {Sp(1)}\}\subset G. \end{aligned}$$
(2.4)

Thus, \(S^2\times S^3\simeq G/H\) is a homogeneous space.

In order to put a metric on \(S^2\times S^3\), Wilking first defines a left invariant metric g on \(G=\mathrm {Sp(1)}\times \mathrm {Sp(1)}\) as follows. Let

$$\begin{aligned} g_0((X,Y),(X',Y'))=\langle X, Y \rangle + \langle X', Y' \rangle , \end{aligned}$$
(2.5)

for \((X,Y),(X',Y')\in \mathfrak {sp}(1)\oplus \mathfrak {sp}(1)=\mathrm {Im}(\mathbb {H})\oplus \mathrm {Im}(\mathbb {H})\), denote a bi-invariant metric. In terms of \(g_0\), the metric g is

$$\begin{aligned} g((X,Y),(X',Y'))=g_0(\Phi (X,Y),(X',Y')), \end{aligned}$$
(2.6)

where \(\Phi \) is a \(g_0\)-symmetric, positive definite endomorphism of \(\mathfrak {sp}(1)\oplus \mathfrak {sp}(1)\) given by

$$\begin{aligned} \Phi = {{\,\mathrm{Id}\,}}- \frac{1}{2}P, \end{aligned}$$
(2.7)

and P is the \(g_0\)-orthogonal projection onto the diagonal subalgebra

$$\begin{aligned} \Delta \mathfrak {sp}(1)\subset \mathfrak {sp}(1)\oplus \mathfrak {sp}(1);\end{aligned}$$
(2.8)

see [11, p. 125].

Wilking’s doubling trick guarantees the existence of a diffeomorphism

$$\begin{aligned} G/H \simeq \Delta G\backslash G\times G/\{1_G\}\times H, \end{aligned}$$
(2.9)

where \( \Delta G\backslash \) denotes the quotient by the left diagonal action of G on \(G \times G\) and H acts on the second factor from the right. Consider the product \((G\times G, g + g)\) (cf. (2.6)) and the induced metric on \(S^2\times S^3 \simeq \Delta G\backslash G\times G/\{1_G\}\times H\) that we denote by \(g_W\). That is, Wilking’s metric \((S^2\times S^3, g_W)\) is the metric that makes the quotient submersion

$$\begin{aligned} (G\times G, g\oplus g) \rightarrow (\Delta G\backslash G\times G/\{1_G\}\times H, g_W) \end{aligned}$$
(2.10)

into a Riemannian submersion. Wilking has shown that \((S^2 \times S^3, g_W)\) has almost-positive curvature, with flat 2-planes located on two hypersurfaces. These hypersurfaces are both diffeomorphic to \(S^2\times S^2\), and they intersect along an \(\mathbb {R}P^3\) [11, Corollary 3, Proposition 6]. However, except for points that lie on four disjoint copies of \(S^2\) inside these two hypersurfaces, there is a unique flat 2-plane. At each point in these four 2-spheres, there is a one parameter family of flat 2-planes and neither the distance curvature nor the biorthogonal curvature of the metric \(g_W\) are strictly positive at any of these points.

3 Proofs

3.1 Proof of Theorem A

We follow Bettiol’s construction of metrics of positive distance curvature on \(S^2 \times S^2\) [2, Theorem], [3, Theorem 6.1], and apply a first-order conformal deformation to Wilking’s metric \((S^2\times S^3, g_W)\) that was described in Section 2.1. This yields metrics of positive distance curvature as in Definition 2, which converge to a metric \(g^0\) as \(\theta \) tends to zero in the \(C^k\)-topology.

Definition 3

Let (M, g) be a compact Riemannian manifold, then, for any function \(\phi :M\rightarrow \mathbb {R}\), and for any small enough \(s>0\), the following is also a Riemannian metric on M

$$\begin{aligned} g_s=(1+s\phi )g, \end{aligned}$$
(3.1)

called the first-order conformal deformation of g.

The variation of sectional curvature of a metric under the first order conformal deformation is given by the following lemma [9]; cf. [3, Chapter 3, Corollary 3.4].

Lemma 1

Let (M, g) be a Riemannian manifold with sectional curvature\(\mathrm {sec}_g\ge 0\), and let \(X,Y\in T_pM\) be g-orthonormal vectors such that \(\mathrm {sec}_g(X\wedge Y)=0\). Consider a first-order conformal deformation \(g_s=(1+s\phi )g\) of g. The first variation of \(\mathrm {sec}_{g_s}(X\wedge Y)\) is

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s}\mathrm {sec}_{g_s}(X\wedge Y)\vert _{s=0}=-\frac{1}{2}\mathrm {Hess}\,\phi (X,X)-\frac{1}{2}\,\mathrm {Hess}\,\phi (Y,Y). \end{aligned}$$
(3.2)

We will also need the following elementary fact [3, Chapter 3, Lemma 3.5].

Lemma 2

Let \(f:[0,S]\times K\rightarrow \mathbb {R}\) be a smooth function, where \(S>0\) and K is a compact subset of a manifold. Assume that \(f(0,x)\ge 0\) for all \(x\in K\), and \(\frac{\partial f}{\partial s}>0\) if \(f(0,x)=0\). Then there exists \(s_*>0\) such that \(f(s,x)>0\) for all \(x\in K\) and \(0<s<s_*\).

Wilking’s metric \((S^2\times S^3, g_W)\) has positive sectional curvature away from a hypersurface Z; see the discussion at the end of Section 2.1. The biorthogonal and distance curvatures are positive inside Z except for points that lie in four disjoint copies of \(S^2\). Every point in these four 2-spheres carries an \(S^1\) worth of flat 2-planes. Denote these four 2-spheres by

$$\begin{aligned} \{S^2_i : i = 1, 2, 3, 4\}.\end{aligned}$$
(3.3)

We only deform Wilking’s metric near these four submanifolds. Let

$$\begin{aligned} \chi _i: S^2\times S^3 \rightarrow \mathbb {R}\end{aligned}$$
(3.4)

denote a bump function of \(S^2_i\), i.e., a nonnegative function that is identically zero outside a tubular neighborhood of \(S^2_i\), and identically one in a smaller tubular neighborhood of \(S^2_i\). Finally, we define four functions

$$\begin{aligned} \{\psi _i:S^2\times S^3 \rightarrow \mathbb {R}: i = 1, 2, 3, 4\}\end{aligned}$$
(3.5)

as

$$\begin{aligned} \psi _i(p)=\mathrm {dist}_{g_W}(p,S^2_i)^2\end{aligned}$$
(3.6)

for \(p\in S^2\times S^3\), where \(\mathrm {dist}_{g_W}\) is the metric distance function on \((S^2\times S^3, g_W)\). Let \(\phi :S^2\times S^3 \rightarrow \mathbb {R}\) be a function defined as

$$\begin{aligned} \phi =-\chi _1\psi _1-\chi _2\psi _2-\chi _3\psi _3-\chi _4\psi _4, \end{aligned}$$
(3.7)

and consider the first-order conformal deformation of \(g_W\) given by

$$\begin{aligned} g_s=(1+s\phi )g_W. \end{aligned}$$
(3.8)

Note that at a point \(p\in S^2_i\), we have

$$\begin{aligned} \mathrm {Hess}\,\phi (X, X)=-\mathrm {Hess}\,\psi _i(X, X)=-2g_W(X_\perp , X_\perp )^2=-2\Vert X_\perp \Vert ^2_{g_W}, \end{aligned}$$
(3.9)

where \(X_\perp \) denotes the component of X perpendicular to \(S^2_i\). For each \(\theta >0\), consider the compact subset of \((S^2\times S^3)\times \mathrm {Gr_2}(T(S^2\times S^3))\times \mathrm {Gr_2}(T(S^2\times S^3))\) given by

$$\begin{aligned} K_\theta :=\{(p,\sigma ,\sigma '):\sigma ,\sigma '\in \mathrm {Gr}_2(T_p(S^2\times S^3)),\mathrm {dist}(\sigma ,\sigma ')\ge \theta \}, \end{aligned}$$
(3.10)

and define

$$\begin{aligned} \begin{aligned} f&:[0,S]\times K_\theta \rightarrow \mathbb {R}, \\ f(s,(p,\sigma ,\sigma '))&:=\frac{1}{2}(\mathrm {sec}_{g_s}(\sigma )+\mathrm {sec}_{g_s}(\sigma ')). \end{aligned} \end{aligned}$$
(3.11)

Notice that \(f(0,(p,\sigma ,\sigma '))\ge 0\) since \(\mathrm {sec}_{g_s}\ge 0\). Furthermore, \(f(0,(p,\sigma ,\sigma '))=0\) only for

$$\begin{aligned} p\in S^2_1\cup S^2_2 \cup S^2_3\cup S^2_4\end{aligned}$$
(3.12)

since these are the only points of \(S^2\times S^3\) that have vanishing biorthogonal and distance curvatures. Let \((p,\sigma ,\sigma ')\) be such that \(f(0,(p,\sigma ,\sigma '))=0\) and let \(\sigma =X\wedge Y\) and \(\sigma '=Z\wedge W\), where X, Y are \(g_W\)-orthonormal, and Z, W are \(g_W\)-orthonormal. Then, by Lemma 1 and equation (3.9), at these points of \(K_\theta \), we have

$$\begin{aligned} \begin{aligned} \frac{\partial f}{\partial s}\vert _{s=0}&=\frac{\mathrm {d}}{\mathrm {d}s}(\mathrm {sec}_{g_s}(X\wedge Y)+\mathrm {sec}_{g_s}(Z\wedge W))\vert _{s=0} \\&=-\frac{1}{2}\mathrm {Hess}\,\phi (X,X)\!-\!\frac{1}{2}\mathrm {Hess}\,\phi (Y,Y)\!-\!\frac{1}{2}\mathrm {Hess}\,\phi (Z,Z)\!-\!\frac{1}{2}\mathrm {Hess}\,\phi (W,W) \\&=\Vert X_\perp \Vert ^2_{g_W}+\Vert Y_\perp \Vert ^2_{g_W}+\Vert Z_\perp \Vert ^2_{g_W}+\Vert W_\perp \Vert ^2_{g_W}>0. \end{aligned} \end{aligned}$$
(3.13)

The previous expression is strictly greater than zero. Indeed, since \(X\wedge Y\) and \(Z\wedge W\) are different 2-planes, \(\mathrm {span}\{X,Y,Z,W\}\) is at least three-dimensional while the submanifolds (3.3) are two-dimensional. Hence, at least one of the perpendicular components \(X_\perp ,Y_\perp ,Z_\perp ,W_\perp \) is nonzero and (3.13) is greater than zero. Since the assumptions of Lemma 2 for the function (3.11) are satisfied, we conclude that there is an \(s_*\) such that \(f(s,(p,\sigma ,\sigma '))>0\) for all \((p,\sigma ,\sigma ')\in K_\theta \) and \(0<s<s_*\). This is precisely the condition \(\mathrm {sec}_{g_s}^\theta >0\) of item (a) of Theorem A. The claims of item (b) and item (c) follow from our construction; cf. [2]. The claim of item (d) follows from [2, Proposition 4.1]. As Bettiol observed in his construction of metrics of positive distance curvature on \(S^2\times S^2\) [2, Section 4.4], for every \(\theta > 0\), there are 2-planes in \((S^2\times S^3, g^{\theta })\) with negative sectional curvature. This completes the proof of Theorem A.\(\square \)

Remark 1

The metrics \((S^2\times S^3, g^\theta )\) of positive distance curvature can be made invariant under the action of certain Deck transformations including the product \(\mathbb {Z}/2\oplus \mathbb {Z}/2\)-action. Indeed, it is possible to perform a local conformal deformation on the orbit space \((\mathbb {R}P^2\times \mathbb {R}P^3, g_W)\) equipped with Wilking’s metric of almost positive curvature, and a similar statement to Theorem A holds for \((\mathbb {R}P^2\times \mathbb {R}P^3, g^\theta )\); cf. [2, Section 4.6].

3.2 Proof of Corollary B

We will use a case of the classification up to diffeomorphism of simply connected 5-manifolds with vanishing second Stiefel–Whitney class due to Smale [8, Theorem A].

Theorem 1

A closed simply connected 5-manifold M with torsion-free homology \(H_2(M; \mathbb {Z}) = \mathbb {Z}^k\) and zero second Stiefel–Whitney class \(w_2(M) = 0\) is determined up to diffeomorphism by its second Betti number \(b_2(M)\). In particular, M is diffeomorphic to a connected sum

$$\begin{aligned} \{S^5\#k(S^2\times S^3): k = b_2(M)\}.\end{aligned}$$
(3.14)

Theorem A and Bettiol’s result regarding the positivity of biorthogonal curvature under connected sums [3, Proposition 7.11] imply that every 5-manifold in the set (3.14) admits a Riemannian metric of positive biorthogonal curvature. \(\square \)

Remark 2

It is natural to ask if the hypothesis \(w_2(M) = 0\) of Corollary B can be removed. Barden has shown that a closed simply connected 5-manifold with torsion-free second homology group is diffeomorphic to a connected sum of copies of \(S^2\times S^3\) and the total space \(S^3\widetilde{\times } S^2\) of the nontrivial 3-sphere bundle over the 2-sphere [1]. It is currently unknown if there is a metric of almost-positive sectional curvature on \(S^3\widetilde{\times } S^2\). Unlike \(S^2\times S^3\), the nontrivial bundle does not arise as a biquotient that satisfies the symmetry hypothesis needed to apply Wilking’s doubling trick; see DeVito’s classification of free circle actions on \(S^3\times S^3\) in [5].

3.3 Proof of Proposition C

The symmetric space metric on \(\mathrm {SU}(3)/\mathrm {SO}(3)\) is the metric that makes the canonical surjection

$$\begin{aligned} \begin{aligned} \pi :\mathrm {SU}(3)&\rightarrow \mathrm {SU}(3)/\mathrm {SO}(3), \\ u&\mapsto u\mathrm {SO}(3), \end{aligned} \end{aligned}$$
(3.15)

into a Riemannian submersion, where \(\mathrm {SU}(3)\) is equipped with a bi-invariant metric. The left action of \(\mathrm {SU}(3)\) on \(\mathrm {SU}(3)/\mathrm {SO}(3)\) induced from the left multiplication on \(\mathrm {SU}(3)\) by (3.15) is transitive and isometric for the symmetric space metric. This means that we can study curvature at one point of \(\mathrm {SU}(3)/\mathrm {SO}(3)\) and isometrically translate the results to any other point. The Cartan decomposition that corresponds to \(\mathrm {SU}(3)/\mathrm {SO}(3)\)

$$\begin{aligned} T_e\mathrm {SU}(3) \simeq \mathfrak {su}(3) = \mathfrak {so}(3) \oplus \mathfrak {so}(3)^\perp \end{aligned}$$
(3.16)

is orthogonal with respect to the bi-invariant metric and it is precisely the decomposition of \(T_e\mathrm {SU}(3)\) into vertical and horizontal subspaces of the Riemannian submersion (3.15). Hence, we have

$$\begin{aligned} T_{ \mathrm {SO}(3) }( \mathrm {SU}(3)/\mathrm {SO}(3) ) \simeq \mathfrak {so}(3)^\perp . \end{aligned}$$
(3.17)

To conclude that \( \mathrm {SU}(3)/\mathrm {SO}(3)\) has positive biorthogonal curvature, we need to show that no two flat 2-planes are orthogonal to each other. A result of Tapp [10, Theorem 1.1] implies that a 2-plane on \( \mathrm {SU}(3)/\mathrm {SO}(3) \) is flat if and only if its horizontal lift is flat. Thus, it is enough to consider horizontal flat 2-planes at the identity of \( \mathrm {SU}(3) \).

A horizontal 2-plane \( X \wedge Y \subset \mathfrak {so}(3)^\perp \) at the identity of \(\mathrm {SU}(3)\) is flat if and only if \( [ X,Y ] = 0 \). Since the maximal number of linearly independent commuting matrices in \(\mathfrak {su}(3)\) is two, every horizontal flat 2-plane corresponds to a maximal abelian subalgebra of \(\mathfrak {so}(3)^\perp \)

$$\begin{aligned} \mathrm {span}_\mathbb {R}\{ X, Y \} = \mathfrak {a}_0 \subset \mathfrak {so}(3)^\perp \,. \end{aligned}$$
(3.18)

By a fundamental fact about the Cartan decomposition, see [7, Proposition 7.29] for the precise statement, any two maximal abelian subalgebras of \(\mathfrak {so}(3)^\perp \) are conjugate by an element of \(\mathrm {SO}(3)\). This means that by fixing one maximal abelian subalgebra, or one horizontal flat 2-plane, we can parametrize all horizontal flat 2-planes by \(\mathrm {SO}(3)\). In what follows, we will obtain an explicit parametrization of horizontal flat 2-planes at the identity of \(\mathrm {SU}(3)\), and so a parametrization of flat 2-planes at a point of \(\mathrm {SU}(3)/\mathrm {SO}(3)\) by choosing a basis for \( \mathfrak {su}(3) \), fixing a horizontal flat 2-plane and parametrizing \( \mathrm {SO}(3) \) by Euler angles. We use this explicit parametrization to show that no two flat 2-planes can be orthogonal. For the basis of \( \mathfrak {su}(3)\), we choose \( \{-i\lambda _i\}_{i=1,...,8} \), where the \( \lambda _i \)’s are traceless, self-adjoint 3 by 3 matrices known as the Gell-Mann matrices [6]. The scalar product on \(\mathfrak {su}(3)\) that corresponds to the bi-invariant metric is

$$\begin{aligned} \langle X, Y \rangle = -\frac{1}{2}\mathrm {Tr}(XY) \end{aligned}$$
(3.19)

for \(X, Y \in \mathfrak {su}(3)\) and the basis \( \{-i\lambda _i\}_{i=1,...,8} \) is orthonormal with respect to (3.19). The Cartan decomposition (3.16) in this basis is

$$\begin{aligned} \mathfrak {so}(3) = \mathrm {span}_\mathbb {R}\{ -i\lambda _2, -i\lambda _5, -i\lambda _7\} \end{aligned}$$
(3.20)

and

$$\begin{aligned} \mathfrak {so}(3)^\perp = \mathrm {span}_\mathbb {R}\{ -i\lambda _1, -i\lambda _3, -i\lambda _4,-i\lambda _6,-i\lambda _8\}. \end{aligned}$$
(3.21)

Matrices \(\lambda _3\) and \(\lambda _8\) are diagonal, so we use \(-\lambda _3 \wedge \lambda _8\) for the reference horizontal flat 2-plane. Every horizontal flat 2-plane, \( X \wedge Y \), with \( X, Y \in \mathfrak {so}(3)^\perp \) such that \( [X,Y]=0 \), can now be written as

$$\begin{aligned} X \wedge Y = -\mathrm {Ad}_r (\lambda _3 \wedge \lambda _8 ) \end{aligned}$$
(3.22)

for some \( r \in \mathrm {SO}(3) \). Suppose that \( X\wedge Y \) and \( X'\wedge Y' \) are two such 2-planes with \(X\wedge Y\) given by (3.22) and \( X' \wedge Y' \) by

$$\begin{aligned} X' \wedge Y' = -\mathrm {Ad}_{r'} (\lambda _3 \wedge \lambda _8 ) \end{aligned}$$
(3.23)

for some \( r' \in \mathrm {SO}(3) \). For the 2-planes (3.22) and (3.23) to be orthogonal, it is necessary and sufficient that the equations

$$\begin{aligned}&\langle \mathrm {Ad}_r \lambda _3, \mathrm {Ad}_{r'} \lambda _3 \rangle = 0, \end{aligned}$$
(3.24)
$$\begin{aligned}&\langle \mathrm {Ad}_r \lambda _3, \mathrm {Ad}_{r'} \lambda _8 \rangle = 0, \end{aligned}$$
(3.25)
$$\begin{aligned}&\langle \mathrm {Ad}_r \lambda _8, \mathrm {Ad}_{r'} \lambda _3 \rangle = 0, \end{aligned}$$
(3.26)

and

$$\begin{aligned} \langle \mathrm {Ad}_r \lambda _8, \mathrm {Ad}_{r'} \lambda _8 \rangle = 0 \end{aligned}$$
(3.27)

hold. Using the \( \mathrm {Ad} \)-invariance of the bi-invariant metric, equations (3.24), (3.25), (3.26), and (3.27) can be rewritten as

$$\begin{aligned}&\langle \lambda _3, \mathrm {Ad}_{r^{-1}r'} \lambda _3 \rangle = 0, \end{aligned}$$
(3.28)
$$\begin{aligned}&\langle \lambda _3, \mathrm {Ad}_{r^{-1}r'} \lambda _8 \rangle = 0, \end{aligned}$$
(3.29)
$$\begin{aligned}&\langle \lambda _8, \mathrm {Ad}_{r^{-1}r'} \lambda _3 \rangle = 0, \end{aligned}$$
(3.30)

and

$$\begin{aligned} \langle \lambda _8, \mathrm {Ad}_{r^{-1}r'} \lambda _8 \rangle = 0. \end{aligned}$$
(3.31)

We now use the Euler angle parametrization of \( \mathrm {SO}(3) \) to write \( r^{-1}r' \in \mathrm {SO}(3) \) as

$$\begin{aligned} r^{-1}r' = \mathrm {exp}(- i \lambda _2 x) \mathrm {exp}(- i \lambda _5 y) \mathrm {exp}(- i \lambda _2 z), \end{aligned}$$
(3.32)

where \( x,y,z \in \mathbb {R} \). Plugging (3.32) into equations (3.28), (3.29), (3.30), and (3.31) and calculating the traces explicitly, we find

$$\begin{aligned}&0 = \langle \lambda _3, \mathrm {Ad}_{r^{-1} r'} \lambda _3 \rangle \nonumber \\&\quad = \frac{1}{4} \mathrm {cos}(2x) \left( 3 + \mathrm {cos}(2y) \right) \mathrm {cos}(2z) - \mathrm {sin}(2x) \mathrm {cos}(y) \mathrm {sin}(2z), \end{aligned}$$
(3.33)
$$\begin{aligned}&0 = \langle \lambda _3, \mathrm {Ad}_{r^{-1} r'} \lambda _8 \rangle = - \frac{\sqrt{3}}{2} \mathrm {cos}(2x)\mathrm {sin}^2(y), \end{aligned}$$
(3.34)
$$\begin{aligned}&0 = \langle \lambda _8, \mathrm {Ad}_{r^{-1} r'} \lambda _3 \rangle = - \frac{\sqrt{3}}{2} \mathrm {cos}(2z)\mathrm {sin}^2(y), \end{aligned}$$
(3.35)

and

$$\begin{aligned} 0 = \langle \lambda _8, \mathrm {Ad}_{r^{-1} r'} \lambda _8 \rangle =\frac{1}{4}(1+3\mathrm {cos}(2y))\,. \end{aligned}$$
(3.36)

Equations (3.34), (3.35), and (3.36) imply \(\mathrm {cos}^2(y)=1/3\) and \(\mathrm {cos}(2x)=\mathrm {cos}(2z)=0\). Plugging this into equation (3.33), we obtain

$$\begin{aligned} \langle \lambda _3, \mathrm {Ad}_{r^{-1} r'} \lambda _3 \rangle \ne 0,\end{aligned}$$
(3.37)

and conclude that there is no solution to the system given by equations (3.33), (3.34), (3.35), and (3.36). This shows that no two 2-flat planes are orthogonal. \(\square \)