• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.47-52
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• 1989

Methods of Riemannian geometry has played an important role in the study of compact transformation groups. Every effective action of a compact Lie group on a differential manifold leaves a Riemannian metric invariant and the study of such actions reduces to the one involving the group of isometries of a Riemannian metric on the manifold which is, a priori, a Lie group under the compact open topology. Once an action of a compact Lie group is given an invariant metric is easily constructed by the averaging method and the Lie group is naturally imbedded in the group of isometries as a Lie subgroup. But usually this invariant metric has more symmetries than those given by the original action. Therefore the first question one may ask is when one can find a Riemannian metric so that the given action coincides with the action of the full group of isometries. This seems to be a difficult question to answer which depends very much on the orbit structure and the group itself. In this paper we give a sufficient condition that a subgroup action of a compact Lie group has an invariant metric which is not invariant under the full action of the group and figure out some aspects of the action and the orbit structure regarding the invariant Riemannian metric. In fact, according to our results, this is possible if there is a larger transformation group, containing the oringnal action and either having larger orbit somewhere or having exactly the same orbit structure but with an orbit on which a Riemannian metric is ivariant under the orginal action of the group and not under that of the larger one. Recently R. Saerens and W. Zame showed that every compact Lie group can be realized as the full group of isometries of Riemannian metric. [SZ] This answers a question closely related to ours but the situation turns out to be quite different in the two problems.

• Journal of the Korean Orthopaedic Association
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• v.56 no.1
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• pp.1-13
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• 2021

Considerable progress on anterior cruciate ligament reconstruction surgery has been made over the past 20 years, and the results have improved significantly. An anatomical understanding of the anterior cruciate ligament has also changed, and the surgical technique has also changed accordingly. The double-bundle concept is still valid, but the ribbon-shaped anterior cruciate ligament concept, including direct fiber and indirect fiber, is gradually replacing it. The isometry point theory no longer exists, and various surgical methods, such as single-bundle anatomical reconstruction, double-bundle reconstruction, remnant preservation, and rectangular tunnel technique, are being performed. Regarding the graft, interest in the bone-patellar tendon-bone, patellar tendon, quadriceps tendon, and allogeneic tendon change over time, and this change is ongoing.

• Journal of the Korean Arthroscopy Society
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• v.12 no.2
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• pp.112-117
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• 2008

Purpose: Double-bundle reconstruction of anterior cruciate ligament(ACL) has the advantage of restoring the isometry and original function of ACL. The purpose of this study is to evaluate the clinical results following double-bundle reconstruction of ACL using autogenous hamstring grafts through an accessory anteromedial portal. Materials and Methods: From January 2005 to July 2006, sixty patients(52 males, 8 females) underwent double-bundle ACL reconstruction using autogenous hamstring tendons..The mean age was 31.7 years($20{\sim}51$ years). The mean follow up period was 13.4 months($12{\sim}16$ months). We made a horizontal-oblique skin incision just medial to tibial tuberosity and harvested semitendinosus and gracilis tendon. Tibial tunnel for posterolateral bundle was made near its anatomical position. By modifying an anatomic reconstruction of ACL by Yasuda et al., we made a femoral tunnel for posterolateral bundle through accessory anteromedial portal. Tunnels for anteromedial bundle were made with conventional method. We reconstructed anteromedial bundle with semitendinosus tendon and posterolateral bundle with gracilis tendon. Clinical results at last follow up were evaluated by range of motion, extent of anterior displacement(KT-1000 arthrometer), pivot-shift test. Functional evaluation of clinical outcomes were evaluated by Lysholm score and modified Feagin Scoring System. Results: There was no limitation of motion of knee joint at last follow up. Mean side to side difference of anterior displacement of tibia by KT-1000 arthrometer was improved from 8.4 mm preoperatively to 1.7 mm postoperatively(p<0.05). Average Lysholm score was improved from 64.1 preoperatively to 92.2 postoperatively(p<0.05). In modified Feagin Scoring System, 90% of cases were rated as good or excellent. Conclusion: Double-bundle reconstruction of ACL using autogenous hamstring grafts through accessory anteromedial portal results in good clinical outcomes.