We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn（n 5） is an odd-dimensional compact submanifold with parallel mean curvature in Sn＋p,and if RicM ＆gt;（n- 2-1n）（1 ＋ H2...We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn（n 5） is an odd-dimensional compact submanifold with parallel mean curvature in Sn＋p,and if RicM ＆gt;（n- 2-1n）（1 ＋ H2） and H ＆lt; δn,where δn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn（n 5） is an odd-dimensional compact submanifold in the space form Fn＋p（c） with c 0,and if RicM ＆gt;（n-2-εn）（c＋H2）,where εn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.展开更多
The present study introduces the carbonatite in the northern part of the Korean Peninsula for the first time.Recent exploration and development of the phosphorus-bearing carbonate rocks in the area have accumulated ne...The present study introduces the carbonatite in the northern part of the Korean Peninsula for the first time.Recent exploration and development of the phosphorus-bearing carbonate rocks in the area have accumulated new geological data which gave us an opportunity to study origin of the carbonate rocks.We conducted geological survey,geochemical analyses of trace elements and rare earth elements,and carbon and oxygen isotope analyses for the carbonatites from Ssangryong,Pungnyon,Yongyu and Puhung districts of the northern part of the Korean Peninsula.This research confirms that the phosphorus-bearing carbonate rocks are carbonatite originating from the mantle.The studied carbonatites are distributed at the junctions of ring and linear structures or around their margins and contain a greater amount of REEs,Y,and Sr than carbonate rocks.The carbonatites in Yongyu and Puhung area show evidence that they were formed from mantle plume generated at the lower mantle and display similar fractionation characteristics to carbonatites in Barrado Itapirapua in Brazil and Kalkfeld and Ondurakorume in Namibia.REE patterns of the carbonatites are typical of carbonatites and the carbon and oxygen isotope analyses demonstrate that the carbonatites were originated from mantle.The carbonatites from the northern part of the Korean Peninsula have a great potential for sources of REE,Y,PGE(platinum group elements),copper,and gold.展开更多
基金This work is supported by Ministry of Science and Technology of China(No.2016YFA0200602 and No.2018YFA0208603)the National Natural Science Foundation of China(No.21573204 and No.21421063)Anhui Initiative in Quantum Information Technologies,Fundamental Research Funds for the Central Universities,National Program for Support of Top-notch Young Professional,CAS Interdisciplinary Innovation Team.
基金supported by National Natural Science Foundation of China (Grant Nos.11071211,11371315 and 11301476)the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of Chinathe China Postdoctoral Science Foundation (Grant No.2012M521156)
文摘We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn（n 5） is an odd-dimensional compact submanifold with parallel mean curvature in Sn＋p,and if RicM ＆gt;（n- 2-1n）（1 ＋ H2） and H ＆lt; δn,where δn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn（n 5） is an odd-dimensional compact submanifold in the space form Fn＋p（c） with c 0,and if RicM ＆gt;（n-2-εn）（c＋H2）,where εn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.
文摘The present study introduces the carbonatite in the northern part of the Korean Peninsula for the first time.Recent exploration and development of the phosphorus-bearing carbonate rocks in the area have accumulated new geological data which gave us an opportunity to study origin of the carbonate rocks.We conducted geological survey,geochemical analyses of trace elements and rare earth elements,and carbon and oxygen isotope analyses for the carbonatites from Ssangryong,Pungnyon,Yongyu and Puhung districts of the northern part of the Korean Peninsula.This research confirms that the phosphorus-bearing carbonate rocks are carbonatite originating from the mantle.The studied carbonatites are distributed at the junctions of ring and linear structures or around their margins and contain a greater amount of REEs,Y,and Sr than carbonate rocks.The carbonatites in Yongyu and Puhung area show evidence that they were formed from mantle plume generated at the lower mantle and display similar fractionation characteristics to carbonatites in Barrado Itapirapua in Brazil and Kalkfeld and Ondurakorume in Namibia.REE patterns of the carbonatites are typical of carbonatites and the carbon and oxygen isotope analyses demonstrate that the carbonatites were originated from mantle.The carbonatites from the northern part of the Korean Peninsula have a great potential for sources of REE,Y,PGE(platinum group elements),copper,and gold.
