数学试题讲解

Let

M

be a closed oriented Riemannian manifold, where

g_{t}

is a family of smooth Riemannian metrics smoothly depending on

t \in (- ε, ε)

. Suppose there exists a family of eigenfunctio ns

f_{t}

and eigenvalues

λ_{t}

smoothly depending on

t

such that

Δ_{g_{t}} f_{t} = λ_{t} f_{t},

where

Δ_{g_{t}}

is the Laplace-Beltrami operator defined using the Riemannian metric

g_{t}

. Additionally, assume that

f_{0}

is not a co nstant function. We define

\dot{λ} := {\frac{d}{d t} |}_{t = 0} λ_{t}

and

\dot{Δ} := {\frac{d}{d t} |}_{t = 0} Δ_{g_{t}}

. Prove the following:
(i) As

λ_{0}

is an eigenvalue of

Δ_{g_{0}}

, let

V_{λ_{0}} := Ker (Δ_{g_{0}} - λ_{0})

be the eigenspace of

λ_{0}

, and let

Π : L^{2} (M, g_{0}) \to V_{λ_{0}}

be th e orthogonal projection onto the eigenspace. Prove that

\dot{λ}

is an eigenvalue of the operator

Π \circ Δ^{'} : V_{λ_{0}} \to V_{λ_{0}}

.
(ii) Let

φ_{t} : M \to M

be a 1-parameter family of diffeomorph isms of

M

and assume

g_{t} = φ_{t}^{*} g_{0}

. Prove that

\dot{λ} = 0