[Paper Review] Curvatures of weighted metrics on tangent sphere bundles

Rui Albuquerque|arXiv (Cornell University)|Jul 26, 2011

Advanced Differential Geometry Research4 citations

TL;DR

This paper derives curvature formulas for weighted metrics on tangent sphere bundles $S_rM$ of a Riemannian manifold $M$, generalizing the Sasaki metric. It establishes that for any $M$ with bounded sectional curvature and fixed radius $r$, positive scalar curvature is achieved on $S_rM$ when the horizontal metric coefficient $f_1$ is sufficiently large or the vertical coefficient $f_2$ is sufficiently small, extending prior results on scalar curvature positivity for tangent sphere bundles.

ABSTRACT

We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional curvature and any chosen constant r.

Motivation & Objective

• To generalize curvature formulas for natural metrics on tangent bundles and radius-$r$ tangent sphere bundles $S_rM$.
• To investigate conditions under which $S_rM$ admits positive scalar curvature metrics.
• To extend known results on scalar curvature positivity for tangent sphere bundles to a broader class of weighted metrics.
• To provide foundational curvature computations toward understanding the holonomy and geometry of the gwistor bundle on $S_1M$ for 4-manifolds.

Proposed method

• Uses a family of weighted metrics $g^{f_1,f_2} = f_1 \pi^*g \oplus f_2 \pi^*g$ on $TM$, with $f_1 = e^{2\varphi_1}$, $f_2 = e^{2\varphi_2}$, and $\varphi_1, \varphi_2$ functions on $M$.
• Applies the Levi-Civita connection of the weighted metric via a modified connection $\nabla^G$ incorporating curvature terms $A$, $B$, and $\nabla^*_{X}Y^v$ with correction for $\varphi_2$.
• Employs the Gauss formula to relate the curvature of $S_rM$ to that of the ambient tangent bundle $TM$, accounting for normal curvature via the second fundamental form.
• Derives explicit expressions for the Ricci and scalar curvatures of $S_rM$ using the ambient curvature and the induced metric structure.
• Utilizes the tensor $\mathcal{R}^\xi(X,Y) = \pi^*R^\nabla(X,Y)\xi$ and its covariant derivatives to compute curvature components.
• Applies the Gauss equation to compute the difference between the Ricci and scalar curvatures of $S_rM$ and the ambient metric $g^{f_1,f_2}$.

Experimental results

Research questions

• RQ1 Under what conditions on the metric coefficients $f_1$ and $f_2$ does the tangent sphere bundle $S_rM$ admit a positive scalar curvature metric?
• RQ2 How do the curvature formulas for weighted metrics on $TM$ extend to the submanifold $S_rM$?
• RQ3 Can the scalar curvature of $S_rM$ be made positive for any Riemannian manifold $M$ with bounded sectional curvature?
• RQ4 What is the role of the function $\varphi_2$ in controlling the scalar curvature of $S_rM$?
• RQ5 How does the curvature of $S_rM$ depend on the radius $r$ and the choice of weighted metric?

Key findings

• For any Riemannian manifold $M$ with bounded sectional curvature and fixed radius $r$, the tangent sphere bundle $S_rM$ admits a positive scalar curvature metric when $f_1$ is sufficiently large or $f_2$ is sufficiently small.
• The scalar curvature of $S_rM$ is given by $\tilde{S}^G = S^G + \frac{(n-1)n}{f_2 r^2}$, where $S^G$ is the scalar curvature of the ambient metric on $TM$, and $n = \dim M - 1$.
• The Ricci curvature of $S_rM$ satisfies $\tilde{\mathrm{ric}}^G = \mathrm{ric}^G + \frac{n-1}{r^2}g|_{V \otimes V}$, showing a correction term from the spherical fibers.
• For surfaces ($\dim M = 2$), the Ricci and scalar curvatures of $TM$ and $S_rM$ coincide under the weighted metric.
• When $f_1$ is constant and $f_2 = e^{2\varphi_2}$, the curvature formulas depend on the covariant derivative of $\varphi_2$ and the curvature tensor of $M$.
• The result generalizes Theorem 1.2 from [6], confirming positive scalar curvature for small $r$ and extending it to a broader class of weighted metrics.