A note about scalar curvature on the total space of a vector bundle

TL;DR

This paper constructs complete Riemannian metrics with positive scalar curvature on the tangent bundles of orientable closed surfaces (excluding tori), addressing a question posed by Gromov.

Contribution

It provides explicit constructions of complete positive scalar curvature metrics on tangent bundles of certain surfaces, extending known results in geometric analysis.

Findings

Tangent bundles of orientable closed surfaces (except tori) admit complete uniformly PSC-metrics.

The construction offers a partial positive answer to Gromov's question on scalar curvature.

The methods may inform future research on scalar curvature on vector bundles.

Abstract

We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.

Equations

$$T_{(p,u)}T\Sigma =V_{(p,U)}T\Sigma \oplus H_{(p,U)}T\Sigma$$

$$\{\begin{array}{l}{g}_{(p,U)}^{a,b}(X^h,Y^h)=g_p(X,Y),g_{(p,U)}^{a,b}(X^h,Y^v)=0\\ g_{(p,U)}^{a,b}(X^v,Y^v)=0.01g_p(X,Y)+[1+\frac{1}{2}{g}_p(U,U)]g_p(X,U)g_p(Y,U)\end{array}$$

$$\{\begin{array}{l}\widetilde{g_{(p,U)}^{a,b}}(X^h,Y^h)=g_p(X,Y),\widetilde{g_{(p,U)}^{a,b}}(X^h,Y^v)=0\\ \widetilde{g_{(p,U)}^{a,b}}(X^v,Y^v)=a(t)g_p(X,Y)+b(t)g_p(X,U)g_p(Y,U)\end{array}$$

$$Sc(\tilde{g})$$

$$F_2=L-M(1+2tL),F_3=N-(M^{\prime }+M^2+2tMN)$$

$$L=\frac{a^{\prime }(t)}{2a(t)},M=\frac{2b(t)-a^{\prime }(t)}{2[a(t)+2tb(t)]}andN=\frac{a(t)b^{\prime }(t)-2a^{\prime }(t)b(t)}{2a(t)[a(t)+2tb(t)]}.$$

$$L=0,M=\frac{1+t}{0.01+2t(1+t)},N=\frac{1}{2[0.01+2t(1+t)]},M^{\prime }=\frac{0.01-2(1+t{)}^2}{[0.01+2t(1+t){]}^2}$$

$$2tMN=\frac{2t(1+t)}{[0.01+2t(1+t){]}^2},F_2=\frac{-(1+t)}{0.01+2t(1+t)},F_3=\frac{(1+t{)}^2-0.005}{[0.01+2t(1+t){]}^2}$$

$$Sc(g^{a,b})(p,U)$$

$$=\frac{7.88t^5+7.76t^4-8.0412t^3-8.0012t^2+3.920197t+1.9998}{[0.01+2t(1+t){]}^2}:=\frac{g(t)}{h(t)}.$$

Full text

The scalar curvature of a Riemannian metric on closed manifolds has been deeply studied (see [RS94], [Ros07],[Sch14] and references therein) for the reason of geometry and general relativity. In this paper, we consider the scalar curvature of a Riemannian metric on the total space of tangent bundle of the orietnable closed surfaces. It is a classic result that any noncompact Riemannian surface admits complete Riemannian metrics with constant negative scalar curvature. Bland and Kalka [BK89] generalized it to manifolds of any dimension: any noncompact manifold of dimensions at least three admits complete Riemannian metrics with constant negative scalar curvature. In contrast to the negative scalar curvature, it exists obstructions to admit a complete Riemannian metric with positive scalar curvature (PSC-metric) on open manifold. If one does not require completeness, then the curvature will not restrict the topology of open manifold according to Gromov’s Theorem [Gro69]. Gromov and Lawson [[GL83], Corollary 6.13] proved that $X\times\mathbf{R}$ carries no complete PSC-metric. Here $X$ is a closed enlargeable manifold [[GL83], Definition 5.5]. For instance, closed Riemannian manifolds with nonpositive sectional curvature are enlargeable manifolds. There is a widely open conjecture: for closed manifold $M^{n}$, $M^{n}\times\mathbf{R}$ admits a complete PSC-metric if and only if $M^{n}$ admits PSC-metrics. Gromov and Lawson also showed in [GL83] that hyperbolic manifold times $\mathbf{R}^{2}$ does not admit complete uniformly PSC-metric. However, if the fiber is the vector space $\mathbf{R}^{n}$($n\geq 3$), the total space of vector bundles (trivial or nontrivial) will admit complete uniformly PSC-metric, as $\mathbf{R}^{n}$ admits $O(n)$-invariant complete uniformly PSC-metrics.

Rosenberg and Stolz [[RS94], proposition 7.2] described a proof that $M^{n}\times\mathbf{R}^{2}$ admits complete PSC-metrics, where $M^{n}$ is a closed manifold. Moreover, Gromov proved the following theorem in [[Gro18], section 4]:

Riemannian manifolds with non-positive sectional curvature are examples of iso-enlargeable manifolds and the definition of the iso-enlargeable manifold can be found in [[Gro18], section 4, p.658]. Gromov posted a question in [[Gro18], section 6]:

Do all products manifold $Y\times\mathbf{R}^{2}$, and, more generally, the total spaces of all $\mathbf{R}^{2}$-bundles admit complete metric $g$ with $Sc(g)\geq 0$, i.e. scalar curvature $\geq 0$? Do, for example, such metrics $g$ exist for compact manifold $Y$ which admit metrics with strictly negative sectional curvature?

Gromov also pointed out in the paper that the best candidates of this kind of manifolds with no complete PSC-metric on them are non-trivial $\mathbf{R}^{2}$-bundles over surfaces of genus at least 2. However, it may has a positive answer.

Let $\Sigma$ be the orientable surface with genus at least 2 which is endowed a hyperbolic metric $g$. Then the Levi-Civita connection $\nabla$ on $\Sigma$ induces a natural splitting of the tangent space $T_{(p,U)}T\Sigma$ into

[TABLE]

Where $(p,u)\in T\Sigma$, $V_{(p,U)}T\Sigma$ is the vertical space, i.e. $Ker\pi_{\ast(p,U)},\pi:T\Sigma\to\Sigma$ and $H_{(p,U)}T\Sigma$ is the horizontal space at $(p,U)$ obtained by using $\nabla$. A horizontal curve in $T\Sigma$ corresponds to a vector field on $\Sigma$ which is parallel with respect to connection $\nabla$ on $(\Sigma,g)$. A vector on $T\Sigma$ is horizontal if it is tangent to a horizontal curve and vertical if it is tangent to a fiber. There exists the horizontal (resp. vertical) distribution $HT\Sigma$ (resp.$VT\Sigma$) and the direct sum decomposition: $TT\Sigma=HT\Sigma\oplus VT\Sigma$. This rises to the horizontal and vertical lifts $X^{h}$, $X^{v}$ of a vector field $X$ on $\Sigma$. The metric $g^{a,b}$ on $T\Sigma$ be constructed by:

[TABLE]

The metric induced by $100\times g^{a,b}$ is no less than the metric induced by Cheeger-Gromoll metric [CG72] which is complete. Thus, it implies the completeness of $g^{a,b}$. Let $t=\frac{1}{2}g_{p}(U,U)$, then the metric $g^{a,b}$ is a special example of the general metric on $T\Sigma$:

[TABLE]

The general metric was firstly defined in [Ana99] and Munteanu computed its scalar curvature [[Mun08], Corollary 2.16]: Let $(M_{C}^{n},g)$ be a space form, then

[TABLE]

where n is the dimension of $M$, $C$ is the constant of sectional curvature of $(M^{n},g)$,

[TABLE]

In putting the data $n=2$, $C=-1$, $a=0.01$, $b=1+t$ to the formulas, one gets

[TABLE]

Then the scalar curvature of $g^{a,b}$ at $(p,U)$ is

[TABLE]

Since $g(t)>1$ for $t\geq 0$ and $g(t)$ increases faster than $h(t)$ when $t$ goes to positive infinity, $Sc(g^{a,b})(p,U)\geq C_{1}>0$ for all $(p,U)\in T\Sigma$. Thus, $g^{a,b}$ is a complete uniformly PSC-metric. ∎

$\mathbf{Acknowledgment}$: I thank Alexander Engel for pointing out the error on my proof of Rosenberg-Stolz Theorem in the first draft, Thomas Schick and Chao Qian for many stimulating conversations, Shuqiang Zhu for proof-read and the funding from China Scholarship Council.