Distance　metric　learning　and　nonlinear　dimensionality　reduction　are　two　interesting　and　active　topics　in　recent　years.　However,　the　connection　between　them　is　not　thoroughly　studied　yet.　In　this　paper,　a　transductive　framework　of　distance　metric　learning　is　proposed　and　its　close　connection　with　many　nonlinear　spectral　dimensionality　reduction　methods　is　elaborated.　Furthermore,　we　prove　a　representer　theorem　for　our　framework,　linking　it　with　function　estimation　in　an　RKHS,　and　making　it　possible　for　generalization　to　unseen　test　samples.　In　our　framework,　it　suffices　to　solve　a　sparse　eigenvalue　problem,　thus　datasets　with　105　samples　can　be　handled.　Finally,　experiment　results　on　synthetic　data,　several　UCI　databases　and　the　MNIST　handwritten　digit　database　are　shown.