The geometry of symmetric triad and orbit spaces of Hermann actions

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

“…4.6, we can apply Ikawa's results (cf. Lemma 4.22 in[10]) to our cases. Let us denote the second fundamental form and the tension field of the orbit K 2 π 1 (x) in N 1 by B H and τ H , respectively.…”

“…However, we can apply our method to the cases of θ 1 ∼ θ 2 . When θ 1 ∼ θ 2 , a Hermann action is orbit equivalent to the isotropy action of a compact symmetric space (see [10]). Hence, it is sufficient to discuss the cases of isotropy actions, that is, θ 1 = θ 2 .…”

“…From this section, we apply the results in Section 3 to the orbits of Hermann actions using symmetric triads (cf. [10], [11], [12]), and determine biharmonic regular orbits of cohomogeneity one Hermann actions. For this purpose, we express the tension field and the square norm of the second fundamental form of orbits of Hermann actions in terms of symmetric triads (Theorem 4.8).…”

Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups

“…For commutative Hermann actions which satisfy one of (A), (B) and (C) in Theorem 3.1, Ikawa obtained the same result (cf. [I1,Theorem 2.24]). Moreover, the first author proved Proposition 3.7, i.e.…”

Examples of austere orbits of the isotropy representations for semisimple pseudo-Riemannian symmetric spaces

“…Type of rank s-rank (R, θ) (e 6(6) , sp(4)) EI 6 6 (e 6(6) , sp(4, R)) EI 6 6 (e 6(6) , sl(6, R) + sl(2, R)) FI 4 4 (e 6(2) , sp(4, R)) EII 6 4 (e 6(6) , sp(2, 2)) EI 6 6 (e 6(6) , so(5, 5) + R) BCI 2 2 (e 6(−14) , sp(2, 2)) EIII 6 2 (e 6(2) , su(6) + su (2)) FI 4 4 (e 6(2) , su(3, 3) + sl(2, R)) FI 4 4 (e 6(2) , su(4, 2) + su (2)) FI 4 4 (e 6(2) , so(6, 4) + so (2)) BCI 2 2 (e 6(−14) , su(4, 2) + su (2)) FIII 4 2 (e 6(−14) , so(10) + so (2)) BCI 2 2 (e 6(−14) , so * (10) + so (2)) BCI 2 2 (e 6(−14) , su(5, 1) + sl(2, R)) FIII 4 2 (e 6(2) , so * (10) + so (2)) BCI 2 2 (e 6(−14) , so(8, 2) + so (2)) BCI 2 2 (e 6(6) , f 4(4) ) AI 2 2 (e 6(−26) , sp(3, 1)) EIV 6 2 (e 6(6) , su * (6) + su (2)) FI 4 4 (e 6(2) , sp(3, 1)) EII 6 4 (e 6(2) , f 4(4) ) AIII 2 1 (e 6(−26) , su * (6) + su (2) (e 7 (7) , su (8)) EV 7 7 (e 7 (7) , sl(8, R)) EV 7 7 (e 7 (7) , su(4, 4)) EV 7 7 (e 7 (7) , so(6, 6) + sl(2, R)) FI 4 4 (e 7(−5) , su(4, 4)) EVI 7 4 (e 7 (7) , su * (8)) EV 7 7 (e 7 (7) , e 6(6) + R) (e 7 (7) , e 6(2) + so (2)) CI 3 3 (e 7(−25) , su(6, 2)) EVII 7 3 (e 7 (7) , so * (12) + su (2)) FI 4 4 (e 7(−5) , su(6, 2)) EVI 7 4 (e 7(−5) , e 6(2) + so (2) (e C 7 , e 7(7) ) EV 7 7 (e 7(7) + e 7 (7) , e 7(7) ) EV 7 7…”