Laplace Operators in Differential Geometry
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.The connection Laplacian is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e, tensors of rank 0), the connection Laplacian is often called the Laplace Beltrami operator. On...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.The connection Laplacian is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e, tensors of rank 0), the connection Laplacian is often called the Laplace Beltrami operator. On a Riemannian manifold, one can define the conformal Laplacian as an operator on smooth functions; it differs from the Laplace Beltrami operator by a term involving the scalar curvature of the underlying metric.
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