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Normal complex contact metric manifolds admitting a semi symmetric metric connection

Aysel Turgut Vanli
Aysel Turgut Vanli
, 
Inan Ünal
Inan Ünal
 and 
Dilek Özdemir
Dilek Özdemir
   | Jul 03, 2020
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Published Online: Jul 03, 2020
Page range: 49 - 66
Received: Sep 01, 2019
Accepted: Oct 11, 2019
DOI: https://doi.org/10.2478/amns.2020.2.00013
Keywords
© 2020 Aysel Turgut Vanli et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

j.amns.2020.2.00013.tab.001.w2aab3b7d276b1b6b1ab1b5c13Aa

R¯(e1,e1∗)e1=2e1∗−e2∗−e2R¯(e1,e1∗)e1∗=−2e1−e2+e2∗R¯(e1,e1∗)e2=−2e2∗+e1+e1∗R¯(e1,e1∗)e2∗=2e2+e1−e1∗R¯(e1,e1∗)U=−2V \matrix{{\overline R ({e_1},e_1^ * ){e_1} = 2e_1^ * - e_2^ * - {e_2}} \cr {\overline R ({e_1},e_1^ * )e_1^ * = - 2{e_1} - {e_2} + e_2^ *} \cr {\overline R ({e_1},e_1^ * ){e_2} = - 2e_2^ * + {e_1} + e_1^ *} \cr {\overline R ({e_1},e_1^ * )e_2^ * = 2{e_2} + {e_1} - e_1^ *} \cr {\overline R ({e_1},e_1^ * )U = - 2V} \cr} R¯(e1,e2)e1=5e2+e1∗R¯(e1,e2)e1∗=−e2∗−e1R¯(e1,e2)e2=−5e1+e2∗R¯(e1,e2)e2∗=e1∗−e2 \matrix{{\overline R ({e_1},{e_2}){e_1} = 5{e_2} + e_1^ *} \cr {\overline R ({e_1},{e_2})e_1^ * = - e_2^ * - {e_1}} \cr {\overline R ({e_1},{e_2}){e_2} = - 5{e_1} + e_2^ *} \cr {\overline R ({e_1},{e_2})e_2^ * = e_1^ * - {e_2}} \cr} R¯(e1∗,e2∗)e1=−e2−e1∗R¯(e1∗,e2∗)e1∗=5e2∗+e1R¯(e1∗,e2∗)e2=e1−e2∗R¯(e1∗,e2∗)e2∗=−5e1∗+e2 \matrix{{\overline R (e_1^ *,e_2^ * ){e_1} = - {e_2} - e_1^ *} \cr {\overline R (e_1^ *,e_2^ * )e_1^ * = 5e_2^ * + {e_1}} \cr {\overline R (e_1^ *,e_2^ * ){e_2} = {e_1} - e_2^ *} \cr {\overline R (e_1^ *,e_2^ * )e_2^ * = - 5e_1^ * + {e_2}} \cr}
R¯(e1∗,e2)e1=e1∗+e2∗R¯(e1∗,e2)e1∗=5e2−e1R¯(e1∗,e2)e2=−5e1∗+e2∗R¯(e1∗,e2)e2∗=−e2−e1 \matrix{{\overline R (e_1^ *,{e_2}){e_1} = e_1^ * + e_2^ *} \cr {\overline R (e_1^ *,{e_2})e_1^ * = 5{e_2} - {e_1}} \cr {\overline R (e_1^ *,{e_2}){e_2} = - 5e_1^ * + e_2^ *} \cr {\overline R (e_1^ *,{e_2})e_2^ * = - {e_2} - {e_1}} \cr} R¯(e1,e2∗)e1=5e2∗+e1∗R¯(e1,e2∗)e1∗=e2−e1R¯(e1,e2∗)e2=e2∗−e1∗R¯(e1,e2∗)e2∗=−5e1−e2 \matrix{{\overline R ({e_1},e_2^ * ){e_1} = 5e_2^ * + e_1^ *} \cr {\overline R ({e_1},e_2^ * )e_1^ * = {e_2} - {e_1}} \cr {\overline R ({e_1},e_2^ * ){e_2} = e_2^ * - e_1^ *} \cr {\overline R ({e_1},e_2^ * )e_2^ * = - 5{e_1} - {e_2}} \cr} R¯(e2,e2∗)e1=−2e1∗−e2−e2∗R¯(e2,e2∗)e1∗=2e1−e2+e2∗R¯(e2,e2∗)e2=e1+e1∗+2e2∗R¯(e2,e2∗)e2∗=e1−2e2∗−e1∗ \matrix{{\overline R ({e_2},e_2^ * ){e_1} = - 2e_1^ * - {e_2} - e_2^ *} \cr {\overline R ({e_2},e_2^ * )e_1^ * = 2{e_1} - {e_2} + e_2^ *} \cr {\overline R ({e_2},e_2^ * ){e_2} = {e_1} + e_1^ * + 2e_2^ *} \cr {\overline R ({e_2},e_2^ * )e_2^ * = {e_1} - 2e_2^ * - e_1^ *} \cr}
R¯(e1,U)V=R¯(e1∗,V)U=e1+e1∗R¯(e2,U)V=R¯(e2∗,V)U=e2+e2∗R¯(e1,V)U=−R¯(e1∗,U)V=e1−e1∗R¯(e2,V)U=−R¯(e2∗,U)V=e2−e2∗ \matrix{{\overline R ({e_1},U)V = \overline R (e_1^ *,V)U = {e_1} + e_1^ *} \cr {\overline R ({e_2},U)V = \overline R (e_2^ *,V)U = {e_2} + e_2^ *} \cr {\overline R ({e_1},V)U = - \overline R (e_1^ *,U)V = {e_1} - e_1^ *} \cr {\overline R ({e_2},V)U = - \overline R (e_2^ *,U)V = {e_2} - e_2^ *} \cr} R¯(e1,e2)U=R¯(e1,e2)V=R¯(e1,e2∗)U=0R¯(e1∗,e2)U=R¯(e1∗,e2)V=R¯(e2∗,e1)U=0R¯(e1∗,e2∗)U=R¯(e1∗,e2∗)V=R¯(e1,e2∗)V=0R¯(e1,e1∗)V=R¯(e2,e2∗)U=2U \matrix{{\overline R ({e_1},{e_2})U = \overline R ({e_1},{e_2})V = \overline R ({e_1},e_2^ * )U = 0} \cr {\overline R (e_1^ *,{e_2})U = \overline R (e_1^ *,{e_2})V = \overline R (e_2^ *,{e_1})U = 0} \cr {\overline R (e_1^ *,e_2^ * )U = \overline R (e_1^ *,e_2^ * )V = \overline R ({e_1},e_2^ * )V = 0} \cr {\overline R ({e_1},e_1^ * )V = \overline R ({e_2},e_2^ * )U = 2U} \cr}
R¯(e2,U)e1=−UR¯(e2,U)e1∗=UR¯(e2,U)e2=−VR¯(e2,U)e2∗=−V \matrix{{\overline R ({e_2},U){e_1} = - U} \cr {\overline R ({e_2},U)e_1^ * = U} \cr {\overline R ({e_2},U){e_2} = - V} \cr {\overline R ({e_2},U)e_2^ * = - V} \cr} R¯(e2,V)e1=−VR¯(e2,V)e1∗=VR¯(e2,V)e2=−UR¯(e2,V)e2∗=U \matrix{{\overline R ({e_2},V){e_1} = - V} \cr {\overline R ({e_2},V)e_1^ * = V} \cr {\overline R ({e_2},V){e_2} = - U} \cr {\overline R ({e_2},V)e_2^ * = U} \cr} R¯(e2∗,U)e1=UR¯(e2∗,U)e1∗=UR¯(e2∗,U)e2=VR¯(e2∗,U)e2∗=−V \matrix{{\overline R (e_2^ *,U){e_1} = U} \cr {\overline R (e_2^ *,U)e_1^ * = U} \cr {\overline R (e_2^ *,U){e_2} = V} \cr {\overline R (e_2^ *,U)e_2^ * = - V} \cr} R¯(e2∗,V)e1=VR¯(e2∗,V)e1∗=VR¯(e2∗,V)e2=−UR¯(e2∗,V)e2∗=−U. \matrix{{\overline R (e_2^ *,V){e_1} = V} \cr {\overline R (e_2^ *,V)e_1^ * = V} \cr {\overline R (e_2^ *,V){e_2} = - U} \cr {\overline R (e_2^ *,V)e_2^ * = - U.} \cr}
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