[Paper Review] Cheeger Gromoll type metrics on the tangent bundle

Marian Ioan Munteanu|ArXiv.org|Sep 30, 2006

Geometry and complex manifolds20 references19 citations

TL;DR

This paper introduces a one-parameter family of Riemannian metrics $g_a$ on the tangent bundle $T(M)$ of a Riemannian manifold $(M,g)$, generalizing the Cheeger-Gromoll metric, and equips $T(M)$ with a compatible almost complex structure $J_a$, making it a locally conformal almost Kähler manifold. The key result is that $T(M)$ with $g_a$ is Kählerian only if $a(t)$ satisfies a specific ODE, and flatness occurs precisely when $a(t) = a_0 \frac{e^{2\sqrt{1+2t}}}{(1+\sqrt{1+2t})^2}$ with $a_0$ chosen so $a(0) = 1$, yielding a flat metric of Cheeger-Gromoll type.

ABSTRACT

In this paper we study a Riemanian metric on the tangent bundle $T(M)$ of a Riemannian manifold $M$ which generalizes the Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to $T(M)$ a structure of locally conformal almost Kählerian manifold. We found conditions under which $T(M)$ is almost Kählerian, locally conformal Kählerian or Kählerian or when $T(M)$ has constant sectional curvature or constant scalar curvature.

Motivation & Objective

• To generalize the Cheeger-Gromoll metric on $T(M)$ via a one-parameter family of metrics $g_a$.
• To define a compatible almost complex structure $J_a$ and study the resulting almost Hermitian structure on $T(M)$.
• To determine conditions under which $(T(M), g_a, J_a)$ is almost Kählerian, locally conformal Kählerian, or Kählerian.
• To investigate when $T(M)$ with $g_a$ has constant sectional curvature or constant scalar curvature.
• To derive a flat metric of Cheeger-Gromoll type on $T(M)$ by solving an ODE for $a(t)$.

Proposed method

• Define a one-parameter family of Riemannian metrics $g_a$ on $T(M)$ that generalize the Cheeger-Gromoll metric by modifying the vertical metric component via a function $a(t)$, where $t = \frac{1}{2}g(\mathtt{u}, \mathtt{u})$.
• Construct a compatible almost complex structure $J_a$ using horizontal and vertical lifts of tangent vectors to $M$, ensuring compatibility with $g_a$.
• Use the orthonormal frame $\{E_i\}$ on $T_{(p,\mathtt{u})}T(M)$ derived from an orthonormal basis of $T_pM$ to compute sectional curvatures $\tilde{K}^a$.
• Derive explicit formulas for sectional curvature $\tilde{K}^a(U,V)$ in terms of the curvature tensor $R$ of $M$ and the function $a(t)$, including components for horizontal, vertical, and mixed planes.
• Compute the scalar curvature $\widetilde{\mathrm{scal}}^a$ of $(T(M), g_a)$ in terms of the scalar curvature of $M$ and curvature contractions involving $R_{e_i e_j} \mathtt{u}$.
• Solve an ODE for $a(t)$: $\frac{a'(t)}{a(t)} = \frac{2}{1 + \sqrt{1 + 2t}}$, leading to the explicit solution $a(t) = a_0 \frac{e^{2\sqrt{1+2t}}}{(1 + \sqrt{1+2t})^2}$, which yields a flat metric when $a_0$ is normalized so $a(0) = 1$.

Experimental results

Research questions

• RQ1 Under what conditions is the manifold $(T(M), g_a, J_a)$ almost Kählerian, locally conformal Kählerian, or Kählerian?
• RQ2 Can $T(M)$ with the metric $g_a$ have constant sectional curvature, and if so, under what conditions on $a(t)$?
• RQ3 For which functions $a(t)$ does $T(M)$ with $g_a$ have constant scalar curvature when $M$ is a real space form?
• RQ4 Is there a non-trivial $a(t)$ such that $(T(M), g_a)$ is flat?
• RQ5 What is the explicit form of the metric $g_a$ that makes $T(M)$ flat, and what is the corresponding $a(t)$?

Key findings

• The manifold $(T(M), g_a)$ is Kählerian only if $a(t)$ satisfies a specific ordinary differential equation, and no such $a(t)$ exists that makes the structure Kählerian, proving that no Cheeger-Gromoll type metric on $T(M)$ yields a Kähler manifold.
• A flat metric of Cheeger-Gromoll type exists on $T(M)$ when $a(t) = a_0 \frac{e^{2\sqrt{1+2t}}}{(1 + \sqrt{1+2t})^2}$ with $a_0$ normalized so $a(0) = 1$, yielding a flat metric $g_1$ on $T(M)$.
• The scalar curvature $\widetilde{\mathrm{scal}}^a$ of $(T(M), g_a)$ is given by a formula involving the scalar curvature of $M$, curvature contractions $|R_{e_i e_j} \mathtt{u}|^2$, and terms in $F_2(t)$, $F_3(t)$, and $a(t)$, with explicit dependence on $m = \dim M$.
• For $M$ a real space form with constant sectional curvature $c$, the condition for $T(M)$ with $g_a$ to have constant scalar curvature leads to a complex ODE involving $a(t)$, $a'(t)$, and $a''(t)$, which is not solvable in elementary functions.
• The sectional curvature $\tilde{K}^a$ of $T(M)$ with $g_a$ is computed explicitly in terms of the curvature tensor of $M$, with components depending on $a(t)$, $R_{e_i e_j} \mathtt{u}$, and $R_{\mathtt{u} e_k} e_i$, showing non-constant curvature in general.
• The metric $g_1$ defined by $a(t) = \frac{4e^{2(\sqrt{1+2t} - 1)}}{(1 + \sqrt{1+2t})^2}$ yields a flat tangent bundle $T(M)$, confirming the existence of a non-trivial flat metric of Cheeger-Gromoll type.