A characterization for totally real submanifolds using self-adjoint differential operator

In this article, we study totally real submanifolds in Kaehler product manifold with constant scalar curvature using self-adjoint differential operator $ \Box $. Under this setup, we obtain a characterization result. Moreover, we discuss $ \delta- $invariant properties of such submanifolds and get an obstruction result as an application of the inequality derived. The results in the article are supported by non-trivial examples.

[1]  An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature , 2018, Journal of Mathematical Analysis and Applications.

[2]  矢野 健太郎,et al.  Structures on manifolds , 1984 .

[3]  Shing-Tung Yau,et al.  Hypersurfaces with constant scalar curvature , 1977 .

[4]  Willmore Submanifolds in a sphere , 2002, math/0210239.

[5]  Bang-Yen Chen,et al.  RIEMANNIAN GEOMETRY OF LAGRANGIAN SUBMANIFOLDS , 2001 .

[6]  Li Haizhong,et al.  Hypersurfaces with constant scalar curvature in space forms , 1996 .

[7]  Masafumi Okumura,et al.  Hypersurfaces and a Pinching Problem on the Second Fundamental Tensor , 1974 .

[8]  Walcy Santos,et al.  SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR IN SPHERES , 1994 .

[9]  N. Alluhaibi,et al.  On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms , 2020 .

[10]  R. Tracinà,et al.  ON A CLASS OF REACTION DIFFUSION SYSTEMS: EQUIVALENCE TRANSFORMATIONS AND SYMMETRIES , 2007 .

[11]  B. Şahin,et al.  Slant submanifolds of Kaehler Product Manifolds , 2007 .

[12]  Jae Won Lee,et al.  Classification of Casorati ideal Lagrangian submanifolds in complex space forms , 2019, Differential Geometry and its Applications.

[13]  Bang-Yen Chen,et al.  Some pinching and classification theorems for minimal submanifolds , 1993 .

[14]  Haizhong Li,et al.  SUBMANIFOLDS WITH CONSTANT SCALAR CURVATURE IN A UNIT SPHERE , 2013 .

[15]  An-Min Li,et al.  An intrinsic rigidity theorem for minimal submanifolds in a sphere , 1992 .

[16]  Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds , 2021 .

[17]  Shiing-Shen Chern,et al.  Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length , 1970 .

[18]  Bang-Yen Chen.  Some new obstructions to minimal and Lagrangian isometric immersions Dedicated to Professor Tadashi Nagano on the occasion of his seventieth birthday , 2000 .

[19]  K. Ogiue,et al.  On totally real submanifolds , 1974 .

[20]  K. Yano,et al.  A totally real surface in ² that is not totally geodesic , 1975 .

[21]  Young Ho Kim,et al.  Totally real submanifolds of a complex space form , 1996 .