论文标题
Submanifolds上拉普拉斯人的高斯公式
The Gauss formulas for Laplacians on submanifolds
论文作者
论文摘要
Riemannian歧管上有几种类型的矢量场的拉普拉斯人。其中包括Bochner和Hodge Laplacian。 Levi-Civita连接的高斯公式将外在连接与内在连接联系起来。我们将连接的高斯公式扩展到对矢量字段的不同类型的vector场上的公式，以在任何codimension $ k \ geq 1 $的子手机上。在此过程中，我们为RICCI操作员提供了一个高斯公式，第二种基本形式的分歧的公式，以及在革命表面上，在谎言衍生物方面的laplacian公式。该公式适用于研究riemannian歧管上不可压缩的Navier-Stokes方程的制定。
There are several types of Laplacians of a vector field on a Riemannian manifold. These include the Bochner and the Hodge Laplacian. The Gauss formula for the Levi-Civita connection relates the extrinsic connection to the intrinsic connection. We extend the Gauss formula for the connection to formulas for the different types of Laplacians of a vector field on a submanifold of any codimension $k\geq 1$. In the process, we derive a Gauss formula for the Ricci operator, formulas for the divergence of the second fundamental form, and a formula for the Laplacian of a $1$-form on a surface of revolution in terms of the Lie derivatives. The formulas have applications to the study of the formulation of the incompressible Navier-Stokes equations on a Riemannian manifold.
