Loxodromes and transformations in pseudo-Hermitian geometry
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In this paper, we prove that a diffeomorphism $f$ on a normal almost contact $3$-manifold $M$ is a CRL-{\it transformation} if and only if $M$ is an $\alpha$-Sasakian manifold. Moreover, we show that a $CR$-loxodrome in an $\alpha$-Sasakian $3$-manifold is a pseudo-Hermitian magnetic curve with a strength $q=\widetilde{r}\eta(\gamma')=(r+\alpha-t)\eta(\gamma')$ for constant $\eta(\gamma')$. A non-geodesic $CR$-loxodrome is a non-Legendre slant helix. Next, we prove that let $M$ be an $\alpha$-Sasakian $3$-manifold such that $(\nabla_Y S)X=0$ for vector fields $Y$ to be orthogonal to $\xi$, then the Ricci tensor $\rho$ satisfies $\rho=2\alpha^2 g$. Moreover, using the CRL-{\it transformation} $\widetilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\widetilde{R}$, the pseudo-Ricci tensor $\widetilde{\rho}$ and the torsion tensor field $ \widetilde{ \mathfrak{T}}^{t} (\widetilde{S}X,Y)$.
Abstract
In this paper, we prove that a diffeomorphism $f$ on a normal almost contact $3$-manifold $M$ is a CRL-{\it transformation} if and only if $M$ is an $\alpha$-Sasakian manifold. Moreover, we show that a $CR$-loxodrome in an $\alpha$-Sasakian $3$-manifold is a pseudo-Hermitian magnetic curve with a strength $q=\widetilde{r}\eta(\gamma')=(r+\alpha-t)\eta(\gamma')$ for constant $\eta(\gamma')$. A non-geodesic $CR$-loxodrome is a non-Legendre slant helix. Next, we prove that let $M$ be an $\alpha$-Sasakian $3$-manifold such that $(\nabla_Y S)X=0$ for vector fields $Y$ to be orthogonal to $\xi$, then the Ricci tensor $\rho$ satisfies $\rho=2\alpha^2 g$. Moreover, using the CRL-{\it transformation} $\widetilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\widetilde{R}$, the pseudo-Ricci tensor $\widetilde{\rho}$ and the torsion tensor field $ \widetilde{ \mathfrak{T}}^{t} (\widetilde{S}X,Y)$.
참고문헌 (13)
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