Geodesic transformations in almost Hermitian geometry
Let \((M,J,g)\) be an almost Hermitian manifold of real even dimension \(n>2\). Let \(B\) be a topologically embedded submanifold of \(M\) with \(\dim B=q\) and let \(\exp_{\nu}\) be the exponential map of the normal bundle \(\nu\) of \(B\). A geodesic transformation \(\varphi_B\) with respect to \(B\) is a local diffeomorphism defined by \(\exp_{\nu}(ru)=p\mapsto\varphi_B(p)=\exp_{\nu}(s(r)u)\); here \(u\) being an arbitrary unit normal vector and \(r\), \(s\) are supposed to be sufficiently small. \(\varphi_B\) leaves \(B\) invariant. Conformal geodesic transformations were studied by the authors in [Math. J. Toyama Univ. 20, 57-77 (1997; Zbl 1076.53510) and Arab. J. Math. Sci. 3, 13-36 (1997; Zbl 0919.53008)]. Let \(N\) be the gradient of the normal distance function and \(\eta\) the 1-form given by \(\eta(X)=g(X,JN)\). A geodesic transformation \(\varphi_B\) is said to be partially conformal if \(\varphi^{\ast}_B g=e^{2\sigma}g+f\eta\otimes\eta\) for some function \(\sigma\), \(f\), where \(f\) depends only on the normal distance function. When \(f=0\), \(\varphi_B\) is conformal. In the paper under review, it is proved that non-isometric partially conformal geodesic transformations exist only when the submanifold \(B\) is a real hypersurface or reduces to a point. The authors find necessary and sufficient conditions for the existence of such transformations in terms of the Jacobi curvature operator. They also study influences of the existence of such transformations on the geometry of the hypersurface and on the ambient space. Complex space forms are characterized among nearly Kähler manifolds with the help of partially conformal transformations.
scientific article; zbMATH DE number 1347385