Take the product $S^2 \times S^2$ of two two-spheres, but perturb the product metric as follows. Think of each $S^2$ as the unit two-sphere in Euclidean 3-space in the standard way so that for $p \in S^2$ the tangent space $T_p S^2 = p^{\perp}$ is the two-plane in 3-space perpindicular to the unit vector p. Then the standard round metric on the unit $S^2$ is given by $<v, v>_p = v \cdot v$ for $v \in T_p S^2$ where the dot product is the standard dot product of 3-space. Now consider $P = (p_1, p_2) \in S^2 \times S^2$ and corresponding $(v_1, v_2) \in T_{p_1} S^2 \times T_{p_2} S^2$ Declare $$|(v_1, v_2) |_p ^2 = A v_1 \cdot v_1 + B v_1 \cdot v_2 + Cv_2 \cdot v_2 $$ for $A, B, C$ constants. This quadratic form is positive definite and so defines a Riemannian metric on the product of the spheres provided (I guess) that
$B^2 < 4AC$ and $A, C > 0$, and in any case, certainly whenever $B$ is small enough relative to $A, C > 0$.

Question 1. Does anyone have a name for this family of metrics on $S^2 \times S^2$? A reference for that name?

When the cross term $B = 0$ this metric is the product metric of two `round' two-spheres whose radii squared are $A$ and $C$ so the isometry group of the metric is the product $O(3) \times O(3)$ and its geodesic flow is integrable. Taking $B \ne 0$ breaks the symmetry to the diagonal $O(3)$ sitting inside the product of $O(3)$'s and it appears (from numerical experiments, expectations, and, perhaps, lack of imagination) that the geodesic flow of the metric is NOT integrable.

Question 2. When $B \ne 0$ can you prove that the geodesic flow is not integrable?

Motivations. These metrics arose in questions from geometic mechanics. The dynamics of a free spherical pendulum for example -- two rods joined by a ``spherical joint'' and thrown into space with no gravity -- can be put into the form of the geodesic equations for such a metric.

Take the product $S^2 \times S^2$ of two two-spheres, but perturb the product metric as follows. Think of each $S^2$ as the unit two-sphere in Euclidean 3-space in the standard way so that for $p \in S^2$ the tangent space $T_p S^2 = p^{\perp}$ is the two-plane in 3-space perpindicular to the unit vector p. Then the standard round metric on the unit $S^2$ is given by $<v, v>_p = v \cdot v$ for $v \in T_p S^2$ where the dot product is the standard dot product of 3-space. Now consider $P = (p_1, p_2) \in S^2 \times S^2$ and corresponding $(v_1, v_2) \in T_{p_1} S^2 \times T_{p_2} S^2$ Declare $$|(v_1, v_2) |_p ^2 = A v_1 \cdot v_1 + B v_1 \cdot v_2 + Cv_2 \cdot v_2 $$ for $A, B, C$ constants. This quadratic form is positive definite and so defines a Riemannian metric on the product of the spheres provided (I guess) that
$B^2 < 4AC$ and $A, C > 0$, and in any case, certainly whenever $B$ is small enough relative to $A, C > 0$.

Question 1. Does anyone have a name for this family of metrics on $S^2 \times S^2$? A reference for that name?

When the cross term $B = 0$ this metric is the product metric of two `round' two-spheres whose radii squared are $A$ and $C$ so the isometry group of the metric is the product $O(3) \times O(3)$ and its geodesic flow is integrable. Taking $B \ne 0$ breaks the symmetry to the diagonal $O(3)$ sitting inside the product of $O(3)$'s and it appears (from numerical experiments, expectations, and, perhaps, lack of imagination) that the geodesic flow of the metric is NOT integrable.

Question 2. When $B \ne 0$ can you prove that the geodesic flow is not integrable?

Motivations. Holger Dullin came up with this family of metrics while we were working on some questions arising in geometic mechanics and symplectic reduction. The dynamics of a free spherical pendulum for example -- two rods joined by a ``spherical joint'' and thrown into space with no gravity -- can be put into the form of the geodesic equations for such a metric.