By using two modified Ricci tensors, we prove some theorems which correspond to Myers's diameter estimate theorem and Bochner's vanishing theorem.

Let (M, g) be a complete and connected Riemannian manifold of dimension n. By using the Bakry-Emery Ricci curvature tensor on M, we prove two theorems which correspond to the Myers compactness theorem.

Let Q be a symmetric bilinear form on R n =R p+q+r with corank r, rank p + q and signature type (p, q), p resp. q denoting positive resp. negative dimensions. We consider the degenerate spin group Spin(Q) = Spin (p, q, r) in the sense of Crumeyrolle and prove that this group is isomorphic to the semi-direct product of the nondegenerate and indefinite spin group Spin(p, q) with the additive matrix group Mat p + q, r . Mathematics Subject Classification (2000). 15A66.Let , be a symmetric bilinear form on R n and let us consider the subspace W of R n W = {w ∈ R n | w, x = 0 for all x ∈ R n }.W is called the radical and the dimension of W is called the co-rank of , . If corank is zero/non-zero, then , is called non-degenerate/degenerate. If W ⊂ R n is any complementary subspace to W , then the restriction of , to W is nondegenerate. If the co-rank of , is r and the the restriction of , to W has signature (p, q) (in the sense that W has an orthogonal decomposition W ⊕ W , where the dimensions of W respectively W are p resp. q and the restriction of , to W resp. W is positive resp. negative definite), then the pair (R n , , ) is said to be of type R p,q,r (see [1], [2]).