Homogeneous Einstein -metrics on compact simple Lie groups and spheres

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• 作者：Zaili Yan^a ; ^{yanzaili@nbu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Shaoqiang Deng^b ; ^{dengsq@nankai.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C21 ; 53C30 ; 53C60
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：147-160
• 全文大小：670 K

文摘

In this paper, we study homogeneous Einstein an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=561ee59c7a09ce05fabc91db207da099" title="Click to view the MathML source">(α,β)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><mrow><mo>(mo><mi>αmi><mo>,mo><mi>βmi><mo>)mo>mrow>math>an>an>an>-metrics on compact Lie groups and spheres. We first show that any left invariant Einstein an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=561ee59c7a09ce05fabc91db207da099" title="Click to view the MathML source">(α,β)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><mrow><mo>(mo><mi>αmi><mo>,mo><mi>βmi><mo>)mo>mrow>math>an>an>an>-metric on a connected compact simple Lie groups except an id="mmlsi4" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=a52e27b0eb814cc3292942ddcb5ca75c">mg class="imgLazyJSB inlineImage" height="13" width="35" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16302255-si4.gif">cript>mg height="13" border="0" style="vertical-align:bottom" width="35" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0362546X16302255-si4.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si4.gif" overflow="scroll"><mstyle mathvariant="normal"><mi>SUmi>mstyle><mrow><mo>(mo><mn>2mn><mo>)mo>mrow>math>an>an>an> with vanishing S-curvature must be a Randers metric. Secondly, we prove that any an id="mmlsi5" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=8c5dcab0fccc57c3880422a206572b28">mg class="imgLazyJSB inlineImage" height="13" width="57" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16302255-si5.gif">cript>mg height="13" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0362546X16302255-si5.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si5.gif" overflow="scroll"><mstyle mathvariant="normal"><mi>Spmi>mstyle><mrow><mo>(mo><mi>nmi><mo>+mo><mn>1mn><mo>)mo>mrow>math>an>an>an>-invariant Einstein an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=561ee59c7a09ce05fabc91db207da099" title="Click to view the MathML source">(α,β)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><mrow><mo>(mo><mi>αmi><mo>,mo><mi>βmi><mo>)mo>mrow>math>an>an>an>-metric on an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si7.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=65d7e30d4c8c6d1539830f07993f323b" title="Click to view the MathML source">S⁴ⁿ⁺³(n∈N⁺)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si7.gif" overflow="scroll"><msup><mrow><mi>Smi>mrow><mrow><mn>4mn><mi>nmi><mo>+mo><mn>3mn>mrow>msup><mrow><mo>(mo><mi>nmi><mo>∈mo><msup><mrow><mi mathvariant="double-struck">Nmi>mrow><mrow><mo>+mo>mrow>msup><mo>)mo>mrow>math>an>an>an> with vanishing S-curvature is either a Randers metric, or an id="mmlsi8" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si8.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=3c2cb8680abe935af97e37d3be05a739">mg class="imgLazyJSB inlineImage" height="13" width="67" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16302255-si8.gif">cript>mg height="13" border="0" style="vertical-align:bottom" width="67" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0362546X16302255-si8.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si8.gif" overflow="scroll"><mstyle mathvariant="normal"><mi>SUmi>mstyle><mrow><mo>(mo><mn>2mn><mi>nmi><mo>+mo><mn>2mn><mo>)mo>mrow>math>an>an>an>-invariant. Finally, we give a complete description of an id="mmlsi9" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si9.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=f59eea0e254f465bd61eeaf569063b45">mg class="imgLazyJSB inlineImage" height="13" width="60" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16302255-si9.gif">cript>mg height="13" border="0" style="vertical-align:bottom" width="60" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0362546X16302255-si9.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si9.gif" overflow="scroll"><mstyle mathvariant="normal"><mi>SUmi>mstyle><mrow><mo>(mo><mi>nmi><mo>+mo><mn>1mn><mo>)mo>mrow>math>an>an>an>-invariant Einstein Finsler metrics on an id="mmlsi10" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302255&_mathId=si10.gif&_user=111111111&_pii=S0362546X16302255&_rdoc=1&_issn=0362546X&md5=ae54773e1f38f8c55a8c8341c0cc4436" title="Click to view the MathML source">S²ⁿ⁺¹(n≥2)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si10.gif" overflow="scroll"><msup><mrow><mi>Smi>mrow><mrow><mn>2mn><mi>nmi><mo>+mo><mn>1mn>mrow>msup><mrow><mo>(mo><mi>nmi><mo>≥mo><mn>2mn><mo>)mo>mrow>math>an>an>an>.