【摘要】 <正> 1 设M是C^r（r>2）级的n维黎曼流形。对于任一点P∈M,存在包含P的坐标邻域（U,φ）,使映射 _φ:U→Rⁿ 是可微同胚。设（y¹,…,yⁿ）是U上局部坐标,a_αβdy^αdy^β是局部坐标系下M的黎曼度更多还原

【Abstract】 Let M be an n-dimensional Riemannian manifold of class C^r（r>2）, and （?）M be a regular oriented domain on M with boundary （?）, and W_{2, 0}²（?）denote the Sobolev space defined on （?） with the property that every u∈W_{2, 0}²（?）vanishes at （?）, i. e., u|_（?）D=0. Let(a_αβ)be the symmetric matrix of the positive definite metric of M, and ▽_a denote the operator of the covariant derivative with respect to a_αβ. For any u∈C^∞（M）, it is convenient to defineu_α=▽_αu, u_αβ=▽_αu_β, u^α=α^αβu_β, u^αβ=α^αγ▽_γu^β. （1≦α, β, γ≦n）.In this paper we establish the followingTheorem. Let u∈W_{2, 0}²（?）. Theninteral from n=（?） (u^αβu_αβdV)=integral from n=（?） （（⊿u）²dV）+integral from n=（?） （Ric（du, du）dV）-integal from n=（?） （（▽_Nu）²Ωds）, where ⊿ is the Laplace-Beltrami operator, Ric （du, du） is the Ricci curvature of M with respect to the vector field du, ▽_N is the directional derivative in the direction of the exterior normal vector N at （?）, Ω is the mean curvature of （?） in M.更多还原