论文标题
在改良的爱因斯坦张量和两个光滑的紧凑型歧管上
On modified Einstein tensors and two smooth invariants of compact manifolds
论文作者
论文摘要
令$（m，g）$为riemannian $ n $ - manifold，我们用$ \ ric $和$ \ scal $ ricci和$ g $的标量曲率表示。对于标量$ k <n $，修改后的爱因斯坦张量表示$ \ eink $定义为$ \ eink：= \ scale \，g -k -k \ ric $。请注意，通常的爱因斯坦张量与$ \ eint $的一半和$ {\ rm ein} _0 = \ scale.g $重合。事实证明，所有这些新的修饰张量（价格为$ 0 <k <n $）仍然是总标量曲率功能的梯度，但就修改的积分标量产品而言。在本文中，我们研究了这些张量的阳性特性，这些特性概括了标量曲率（$ k = 0 $）和阳性爱因斯坦曲率（$ k = 2 $）的阳性特性。 $ \ eink $的某些正$ k $的积极性意味着所有$ {\ rm ein} _l $带有$ 0 \ leq l \ leq k $的积极性，因此我们定义了一个平稳的不变性$ \ cein（m）$ m $的$ m $ $ m $的$ \ m $，是正面K的正面K $ $ \ eink $ \ eink $ \ eink。从定义上讲，$ \ cein（m）\在[0，n] $中，仅当$ m $没有正面标量曲率指标时，它是零，如果$ m $具有带有正标曲率的爱因斯坦度量，则最大等于$ n $。从某种意义上说，$ \ cein（m）$衡量$ m $要承认爱因斯坦的积极标态曲率。在本文中，我们证明$ \ cein（m）\ geq 2 $如果$ m $承认非Abelian Connected Lie Group的有效操作，或者仅通过正面标量曲率和尺寸$ \ geq 5 $连接。我们也证明，在歧管$ m $上进行手术操作或假设歧管$ m $具有较高的连接性后，不变的$ \ cein $增加。我们证明条件$ \ cein（m）\ leq n-2 $并不意味着对$ m $的第一个基本组有任何限制。我们定义并证明了类似不变的$ \ cein（m）$的类似属性。该论文包含几个开放问题。
Let $(M,g)$ be a Riemannian $n$-manifold, we denote by $\Ric$ and $\Scal$ the Ricci and the scalar curvatures of $g$. For scalars $k<n$, the modified Einstein tensors denoted $\Eink$ are defined as $\Eink :=\Scal \, g -k\Ric$. Note that the usual Einstein tensor coincides with the half of $\Eint$ and ${\rm Ein}_0=\Scal.g$. It turns out that all these new modified tensors, for $0<k<n$, are still gradients of the total scalar curvature functional but with respect to modified integral scalar products. In this paper we study the positivity properties of these tensors that generalize the positivity properties of the scalar curvature ($k=0$) and positive Einstein curvature ($k=2$). The positivity of $\Eink$ for some positive $k$ implies the positivity of all ${\rm Ein}_l$ with $0\leq l\leq k$ and so we define a smooth invariant $\cEin(M)$ of $M$ to be the supremum of positive k's that renders $\Eink$ positive. By definition $\cEin(M)\in [0,n]$, it is zero if and only if $M$ has no positive scalar curvature metrics and it is maximal equal to $n$ if $M$ possesses an Einstein metric with positive scalar curvature. In some sense, $\cEin(M)$ measures how far is $M$ to admit an Einstein metric of positive scalar curvature. In this paper we prove that $\cEin(M)\geq 2$ if $M$ admits an effective action by a non abelian connected Lie group or if $M$ is simply connected of positive scalar curvature and dimension $\geq 5$. We prove as well that the invariant $\cEin$ increases after a surgery operation on the manifold $M$ or by assuming that the manifold $M$ has higher connectivity. We prove that the condition $\cEin(M)\leq n-2$ does not imply any restriction on the first fundamental group of $M$. We define and prove similar properties for an analogous invariant namely $\cein(M)$. The paper contains several open questions.
