Let
M be a
Riemannian manifold with dimension
n, and let
BM(
p,
r) be a geodesic ball centered at
p with radius
r less than the
injectivity radius of
p ∈
M. For each real number
k, let
N(
k) denote the
simply connected space form of dimension
n and constant
sectional curvature k. Cheng's eigenvalue comparison theorem compares the first eigenvalue λ1(
BM(
p,
r)) of the Dirichlet problem in
BM(
p,
r) with the first eigenvalue in
BN(
k)(
r) for suitable values of
k. There are two parts to the theorem: • Suppose that
KM, the
sectional curvature of
M, satisfies ::K_M\le k. :Then ::\lambda_1\left(B_{N(k)}(r)\right) \le \lambda_1\left(B_M(p,r)\right). The second part is a comparison theorem for the
Ricci curvature of
M: • Suppose that the Ricci curvature of
M satisfies, for every vector field
X, ::\operatorname{Ric}(X,X) \ge k(n-1)|X|^2. :Then, with the same notation as above, ::\lambda_1\left(B_{N(k)}(r)\right) \ge \lambda_1\left(B_M(p,r)\right). S.Y. Cheng used
Barta's theorem to derive the eigenvalue comparison theorem. As a special case, if
k = −1 and inj(
p) = ∞, Cheng’s inequality becomes
λ*(
N) ≥
λ*(
H n(−1)) which is
McKean’s inequality. ==See also==