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Let π : Z → Y be a Riemannian V -submersion of compact Vmanifolds. We study when the pull-back of an eigenform of the Laplacian on Y is an eigenform of the Laplacian on Z, and when the associated eigenvalue can change.2000 Mathematics Subject Classification. Primary 58J50.

Earthquakes have occurred globally over the last two decades, which have resulted in an increased expectation of acceptable performance for bridge structures during seismic events. Many research projects have been performed to assess the degree of structural damage especially upper structure of bridges. However, a few researches are available on the pile foundation systems and their response to earthquakes especially for a pile to pile-cap connection. When a pile-cap is loaded, moment and axial loads are transferred to the pile through the pile to pile-cap connection. There is uncertainly with regards the portion of the loads the connection experiences due to the inability to effectively monitor the forces the connection experiences during loadings. In order to investigate behavior of the pile to pile-cap connection for steel pipe piles subject to axial and uplift loads, full scale tests were conducted, and their key factors were considered from test results.