乘积流形的巴赫流
Bach flows of product manifolds
摘要：
我们调查使用巴赫张量定义的几何流的各个方面。首先，我们利用一个著名的分裂的Bach张量分量的$(2,2)$无翘曲的产品流形，我们解决了Bach流方程的典型例子的产品流形，如$S^2\times S^2$，$R^2\times S^2$。此外，我们还得到了一般$(2,2)$流形的不动点条件，并在一个限制情形下求解了它。接下来，我们考虑弯曲流形。对于一类特殊的非对称弯曲四维流形上的Bach流，我们将流方程化为一阶动力系统，并精确求解该动力系统以得到流的特性。比较了Bach流和Ricci流的结果，并定性地讨论了它们之间的差别。最后，我们最后提到了未来工作的可能方向。
英文摘要：
We investigate various aspects of a geometric flow defined using the Bach tensor. Firstly, using a well-known split of the Bach tensor components for $(2,2)$ unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like $S^2\times S^2$, $R^2\times S^2$. In addition, we obtain the fixed point condition for general $(2,2)$ manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped four manifolds, we reduce the flow equations to a first order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.
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