Geodesic
CHARACTERIZATIONS OF RICCI FLAT METRICS AND LAGRANGIAN SUBMANIFOLDS IN TERMS OF THE VARIATIONAL PROBLEM
Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 643-651

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DOI

Given the pair (P, η) of (0,2) tensors, where η defines a volume element, we consider a new variational problem varying η only. We then have Einstein metrics and slant immersions as critical points of the 1st variation. We may characterize Ricci flat metrics and Lagrangian submanifolds as stable critical points of our variational problem. 
TANIGUCHI, TETSUYA; UDAGAWA, SEIICHI. CHARACTERIZATIONS OF RICCI FLAT METRICS AND LAGRANGIAN SUBMANIFOLDS IN TERMS OF THE VARIATIONAL PROBLEM. Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 643-651. doi: 10.1017/S0017089514000536
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     author = {TANIGUCHI, TETSUYA and UDAGAWA, SEIICHI},
     title = {CHARACTERIZATIONS {OF} {RICCI} {FLAT} {METRICS} {AND} {LAGRANGIAN} {SUBMANIFOLDS} {IN} {TERMS} {OF} {THE} {VARIATIONAL} {PROBLEM}},
     journal = {Glasgow mathematical journal},
     pages = {643--651},
     year = {2015},
     volume = {57},
     number = {3},
     doi = {10.1017/S0017089514000536},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000536/}
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