An $SO(3)\times SO(8)$-invariant Einstein metric on $S^3\times S^7$

Abstract: In this paper, we prove the existence of an $SO(3)\times SO(8)$-invariant Einstein metric with positive scalar curvature on $S^{3}\times S<sup>7$.

• The paper demonstrates the existence of a non-standard SO(3)×SO(8)-invariant Einstein metric on S³×S⁷ by leveraging precise winding estimates of Ricci flat trajectories.
• It employs a detailed cohomogeneity one ODE analysis to track invariant metric trajectories and their intersection properties in phase space.
• Numerical and analytical evidence confirms that the excess winding in the Ricci flat trajectory ensures the additional intersection necessary for constructing the novel metric.

An $SO(3)\times SO(8)$-Invariant Non-Standard Einstein Metric on $S^3\times S^7$

Overview

The paper establishes the existence of a non-standard Einstein metric with positive scalar curvature on $S^3\times S^7$ that is invariant under the $SO(3)\times SO(8)$ action (2606.08888). Despite the abundance of product Einstein metrics on products of spheres, finding invariant but non-product (non-standard) Einstein metrics remains a central pursuit in Riemannian geometry. The cornerstone of the construction is a detailed analysis of the cohomogeneity one Einstein ODEs and the winding behavior of Böhms's Ricci flat trajectories in invariant metric spaces, culminating in new existence results for $S^3 \times S^7$.

Context and Motivation

While canonical product Einstein metrics are trivial to construct for $S^3\times S^7$, non-standard homogeneous and cohomogeneity one Einstein metrics are exceptional. Previous breakthroughs—Jensen's homogeneous Einstein metrics, Böhms's cohomogeneity one Einstein metrics for $d_1+d_2\leq 8$, and the nearly Kähler metric on $S^3\times S^3$—serve as important precedents. The method for analyzing invariant Einstein metrics leverages reductions to ODEs via symmetry (cohomogeneity one), where geometric and topological features are encoded in the dynamical properties of the system.

Notably, for dimensions $d_1+d_2=9$, Nienhaus-Wink found non-round Einstein metrics on $S^{10}$ using estimates on the trajectory of Böhms's Ricci flat metric in the phase portrait of the Einstein ODEs. Inspired by these techniques, this work isolates a case—$S^3\times S^7$0—where the winding behavior is particularly pronounced, allowing for the detection of a novel invariant metric on $S^3\times S^7$1.

Formulation of the Einstein ODEs and Symmetry Structure

An Einstein metric on a manifold $S^3\times S^7$2 satisfies $S^3\times S^7$3 for some real $S^3\times S^7$4. On $S^3\times S^7$5 with $S^3\times S^7$6-invariant metrics, the geometry is determined by two warping functions and the system can be re-expressed using coordinates $S^3\times S^7$7 adapted from recent advances [EinsteinMetricsOnTheTenSpheres]. These coordinates facilitate the ODE analysis and compactification of the solution space.

A crucial geometric feature is that the phase space $S^3\times S^7$8, parametrized by $S^3\times S^7$9, is compact and invariant, ensuring global existence for solutions. The variable $S^3\times S^7$0 (a normalized mean curvature) is strictly decreasing in the relevant region, and various fixed points correspond to significant geometric collapses (singular orbits and cone solutions).

The challenge is to characterize entire trajectories emanating from a fixed point corresponding to the collapse of $S^3\times S^7$1, and to track their behavior up to where $S^3\times S^7$2 may collapse—a process corresponding to the existence of a smooth Einstein metric on the double sphere product.

The Winding Analysis and Key Numerical Findings

The novelty of this work resides in a precise winding estimate for Böhms's Ricci flat trajectory $S^3\times S^7$3 within the $S^3\times S^7$4-plane, particularly for the critical case $S^3\times S^7$5. The core result is the measurement of the winding angle as $S^3\times S^7$6, which is substantially larger than for other dimension splits where $S^3\times S^7$7 but $S^3\times S^7$8. This surplus winding is essential to ensure that the unstable manifold from the initial singular orbit loops far enough in phase space to permit additional intersections corresponding to non-standard Einstein metrics.

Figure 1: Distinctive winding of the Ricci flat trajectory for $S^3\times S^7$9 enables the construction of a non-standard invariant Einstein metric.

By comparison, other configurations with $SO(3)\times SO(8)$0, such as $SO(3)\times SO(8)$1, etc., exhibit insufficient winding, as visualized in Figure 2. This difference is not merely quantitative but underpins the qualitative emergence (or failure) of new geometric structures.

Figure 2: In cases with $SO(3)\times SO(8)$2, $SO(3)\times SO(8)$3, the winding is insufficient to produce additional intersections in the moduli space.

Through a combination of analytical and numerical methods, the author rigorously constructs a forward invariant set $SO(3)\times SO(8)$4 in the phase space, trapping the Ricci flat trajectory and allowing sharp control over its terminal angle as it approaches the cone solution. The construction of $SO(3)\times SO(8)$5 (see Figure 3) is critical for establishing the robustness of the winding estimate.

Figure 3: The forward-invariant set $SO(3)\times SO(8)$6 in the $SO(3)\times SO(8)$7 plane, which prevents escape of trajectories and guarantees a precise winding behavior.

Main Theorem and Existence Argument

The central theorem is the existence of a non-standard $SO(3)\times SO(8)$8-invariant Einstein metric with positive scalar curvature on $SO(3)\times SO(8)$9. The existence is reduced to demonstrating that the curve (unstable manifold) in phase space parametrizing Einstein metrics intersects the locus corresponding to smooth collapse of both singular orbits more than once: once for the product metric, and once for a genuinely non-standard metric. The excess winding for $S^3 \times S^7$0 ensures that the curve passes through the region of interest twice.

A key technical device is a carefully designed angle barrier function, leveraging linearization along the cone solution and the construction of explicit "barrier" ODEs to bound the approach angle to the fixed point. This, combined with monotonicity and symmetry properties, implements a shooting argument: if the terminal angle exceeds $S^3 \times S^7$1, the intermediate value theorem guarantees an additional intersection corresponding to a non-standard Einstein metric.

Numerical and Analytical Evidence

Strong corroboration comes from both the phase portrait windings and auxiliary algebraic lemmas verifying forward invariance of prescribed sets. The analysis is structurally robust due to hyperbolicity of the fixed points and the compactness of $S^3 \times S^7$2, which preclude pathological trajectory escape scenarios.

Further, the argument unifies and generalizes earlier existence arguments for non-standard Einstein metrics on sphere products and links them tightly to the geometric analysis of cohomogeneity one Ricci flat manifolds.

Implications and Future Directions

This work contributes a new example of a rigid, non-standard Einstein metric with high symmetry, directly relevant for geometric analysis, global differential geometry, and the theory of special holonomy. The explicit construction in dimension 10 suggests that winding-based analysis could yield additional non-product Einstein metrics in higher dimensions, particularly as numerical and computer-assisted analytic techniques become more powerful.

Practically, the result has implications for the landscape of Einstein metrics on homogeneous spaces, with potential repercussions for string theory, global analysis, and the construction of Ricci solitons or more exotic geometric flows.

On a theoretical level, it raises further questions regarding necessary and sufficient conditions for the existence of non-standard invariant Einstein metrics across arbitrary product manifolds and suggests that topological or representation-theoretic features (encoded via winding phenomena) may provide general criteria.

Conclusion

The paper conclusively demonstrates, via meticulous analysis of phase space windings in the Einstein ODEs and careful barrier construction, the existence of an $S^3 \times S^7$3-invariant, non-standard Einstein metric with positive scalar curvature on $S^3 \times S^7$4. The result bridges geometric intuition and rigorous dynamical analysis, expands the known catalog of such metrics, and sets the stage for further exploration of Einstein geometry on highly symmetric manifolds.