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Journal ArticleDOI

Integral formulas in riemannian geometry

01 Mar 1973-Bulletin of The London Mathematical Society (Oxford University Press (OUP))-Vol. 5, Iss: 1, pp 124-125
About: This article is published in Bulletin of The London Mathematical Society.The article was published on 1973-03-01. It has received 331 citations till now. The article focuses on the topics: Riemannian geometry & Fundamental theorem of Riemannian geometry.
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TL;DR: In this paper, an obstruction for hypersymplectic manifolds equipped with a free, isometric action of SU(1,1) was investigated, and it was shown that the manifold is a metric cone over a split 3-Sasakian manifold.
Abstract: We investigate an obstruction for hypersymplectic manifolds equipped with a free, isometric action of SU(1,1). When the obstruction vanishes, we show that the manifold is a metric cone over a split 3-Sasakian manifold. Furthermore, if the action of SU(1,1) is also proper, then the hypersymplectic manifold fibres over a para-quaternionic Kahler manifold. We conclude the article with some examples for which the obstruction vanishes. In particular, we show that the moduli space to Nahm-Schmid equations admits a fibration over a para-quaternionic Kahler manifold.

1 citations

01 Jan 2004
TL;DR: In this paper, a skew symetric conformal vector fleld on a closed concircular almost contact manifold was considered and its properties were investigated under the assumption that a skew symmetric Killing vector is tangent to the invariant submanifold.
Abstract: We consider a skew symetric conformal vector fleld on a closed concircular almost contact manifold M and flnd its properties. Also, for a CR-product submanifold M 0 of M, the mean curvature vector fleld of the invariant submanifold and the ∞atness of the antiinvariant submanifold are studied, under the assumption that M 0 admits a skew symmetric Killing vector fleld tangent to the invariant submanifold, such that its generative is tangent to the antiinvariant submanifold.

1 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that if a 3D para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature $-1$ or it reduces to a gradient Ricci Soliton.
Abstract: The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if $(g, V,\lambda)$ be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold $M$, then it reduces to a Ricci soliton and the soliton is expanding for $\lambda$=-2. Besides these, in this section, we prove that if $V$ is pointwise collinear with $\xi$, then $V$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature $-1$ or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.

1 citations