On C-totally real minimal submanifolds of the Sasakian space forms with parallel Ricci tensor
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas - Tập 116 - Trang 1-25 - 2022
Tóm tắt
Cheng et al. recently (Results Math 76:144, 2021) established a complete classification of the n-dimensional C-totally real minimal submanifolds with constant sectional curvature in the
$$(2n+1)$$
-dimensional Sasakian space form
$$N^{2n+1}(c)$$
. In this paper, trying to extend the above result, we classify C-totally real minimal submanifolds in
$$N^{2n+1}(c)$$
with parallel Ricci tensor for
$$n=3,4$$
. In particular, we show that 4-dimensional C-totally real minimal Einstein submanifolds in
$$N^9(c)$$
are of constant sectional curvature.
Tài liệu tham khảo
Antić, M., Li, H., Vrancken, L., Wang, X.: Affine hypersurfaces with constant sectional curvature. Pac. J. Math. 310, 275–302 (2021)
Baikoussis, C., Blair, D.E., Koufogiorgos, T.: Integral submanifolds of Sasakian space forms \(M^7(k)\). Results Math. 27, 207–226 (1995)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Birkhäuser, Boston (2010)
Blair, D.E., Carriazo, A.: The contact Whitney sphere. Note Mat. 20(2000/01), 125–133 (2002)
Cheng, X., He, H., Hu, Z.: \(C\)-totally real submanifolds with constant sectional curvature in the Sasakian space forms. Results Math. 76, 144 (2021)
Cheng, X., Hu, Z., Li, A.-M., Li, H.: On the isolation phenomena of Einstein manifolds-submanifolds versions. Proc. Am. Math. Soc. 146, 1731–1740 (2018)
Cheng, X., Hu, Z., Moruz, M., Vrancken, L.: On product affine hyperspheres in \({\mathbb{R} }^{n+1}\). Sci. China Math. 63, 2055–2078 (2020)
Cheng, X., Hu, Z., Moruz, M., Vrancken, L.: On product minimal Lagrangian submanifolds in complex space forms. J. Geom. Anal. 31, 1934–1964 (2021)
Dillen, F., Vrancken, L.: \(C\)-totally real submanifolds of \({\mathbb{S} }^7(1)\) with nonnegative sectional curvature. Math. J. Okayama Univ. 31, 227–242 (1989)
Ejiri, N.: Totally real minimal immersions of \(n\)-dimensional real space forms into \(n\)-dimensional complex space forms. Proc. Am. Math. Soc. 84, 243–246 (1982)
Fetcu, D., Oniciuc, C.: Biharmonic integral \({\cal{C} }\)-parallel submanifolds in \(7\)-dimensional Sasakian space forms. Tohoku Math. J. 64, 195–222 (2012)
Hu, Z., Li, H., Vrancken, L.: On four-dimensional Einstein affine hyperspheres. Differ. Geom. Appl. 50, 20–33 (2017)
Hu, Z., Xing, C.: On the Ricci curvature of \(3\)-submanifolds in the unit sphere. Arch. Math. (Basel) 115, 727–735 (2020)
Hu, Z., Xing, C.: New characterizations of the Whitney spheres and the contact Whitney spheres. Mediterr. J. Math. 19, 75 (2022)
Hu, Z., Yin, J.: An optimal inequality related to characterizations of the contact Whitney spheres in Sasakian space forms. J. Geom. Anal. 30, 3373–3397 (2020)
Lee J.W., Lee C.W., Vîlcu G.-E.: Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms. J. Geom. Phys. 155, 103768 (2020) [II, 171, 104410 (2022)]
Li, H., Wang, X.: Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space. Results Math. 59, 453–470 (2011)
Luo, Y.: Contact stationary Legendrian surfaces in \({\mathbb{S} }^5\). Pac. J. Math. 293, 101–120 (2018)
Luo, Y., Sun, L.: Complete Willmore Legendrian surfaces in \({\mathbb{S} }^5\) are minimal Legendrian surfaces. Ann. Global Anal. Geom. 58, 177–189 (2020)
Mihai, I.: Ideal \(C\)-totally real submanifolds in Sasakian space forms. Ann. Mat. Pura Appl. 182, 345–355 (2003)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Pitiş, G.: Integral submanifolds with closed conformal vector field in Sasakian manifolds. N. Y. J. Math. 11, 157–170 (2005)
Reckziegel, H.: A correspondence between horizontal submanifolds of Sasakian manifolds and totally real submanifolds of Kählerian manifolds. Topics in Differential Geometry, vols. I, II (Debrecen, 1984), pp. 1063–1081. Colloq. Math. Soc., North-Holland (1988)
Sasahara, T.: A class of biminimal Legendrian submanifolds in Sasakian space forms. Math. Nachr. 287, 79–90 (2014)
Sasahara, T.: Classification of biharmonic \({\cal{C} }\)-parallel Legendrian submanifolds in \(7\)-dimensional Sasakian space forms. Tohoku Math. J. 71, 157–169 (2019)
Szabó, Z.I.: Structure theorems on Riemannian spaces satisfying \(R(X,Y)\cdot R=0\). I. The local version. J. Differ. Geom. 17, 531–582 (1982)
Tanno, S.: Sasakian manifolds with constant \(\phi \)-holomorphic sectional curvature. Tohoku Math. J. 21, 501–507 (1969)
Verheyen, P., Verstraelen, L.: Conformally flat \(C\)-totally real submanifolds of Sasakian space forms. Geom. Dedicata 12, 163–169 (1982)
Verstraelen, L., Vrancken, L.: Pinching theorems for \(C\)-totally real submanifolds of Sasakian space forms. J. Geom. 33, 172–184 (1988)
Yamaguchi, S., Kon, M., Ikawa, T.: \(C\)-totally real submanifolds. J. Differ. Geom. 11, 59–64 (1976)