It is shown that the Wahlquist metric, which is a stationary, axially symmetric perfect fluid solution with ρ + 3p = const, admits a rank-2 generalized closed conformal Killing-Yano tensor with a skew-symmetric torsion. Taking advantage of the presence of such a tensor, we obtain a higherdimensional generalization of the Wahlquist metric in arbitrary dimensions, including a family of vacuum black hole solutions with spherical horizon topology such as Schwarzschild-Tangherlini, Myers-Perry and higher-dimensional Kerr-NUT-(A)dS metrics and a family of static, spherically symmetric perfect fluid solutions in higher dimensions.

A supersymmetric solution to type II supergravity is constructed by superposing two hyperKählers with torsion metrics. The solution is given by a Kähler with torsion metric with SU (3) holonomy. The metric is embedded into a heterotic solution obeying the Strominger system, together with a Yang-Mills instanton obtained by the standard embedding. T dualities lead to an SO(6) instanton describing a symmetry breaking from E8 to SO(10). The compactification by taking a periodic array yields a supersymmetric domain wall solution of heterotic supergravity.

We study supersymmetric solutions in 7-and 8-dimensional Abelian heterotic supergravity theories. In dimension 7, the solutions are described by G 2 with torsion equations. When a G 2 manifold has principal orbits S 3 × S 3 , the equations are reduced to ordinary differential equations with four radial functions. For these equations we obtain explicit ALC metrics with S 3 -bolt and T 1,1 -bolt singularities. In dimension 8, we study supersymmetric solutions to Spin(7) with torsion equations associated with 3-Sasakian manifolds by using a similar method to the case G 2 . I. INTRODUCTIONThe supersymmetric equations of supergravity theory are expressed as Killing spinor equations.Specifically, the geometry of a Neveu-Schwarz(NS-NS) sector in type II supergravity theory and an NS sector in heterotic supergravity theory can be described in terms of G-structures. The use of G-structures rehashes the supersymmetric equations as equations rewritten by means of differential forms. A G-structure has provided us with a technique for constructing supersymmetric solutions for these theories[1][2].In this paper, we study supersymmetric solutions in 7-and 8-dimensional Abelian heterotic supergravity theories, where a G 2 -structure on 7-dimensional manifolds and a Spin(7)-structure on 8-dimensional manifolds play an important role. The space of intrinsic torsion can be decomposed into the irreducible components associated with representations of G 2 or Spin(7). The classification of G 2 -or Spin(7)-structures is given by all possible combinations of these irreducible components.The different classes of these structures are characterized via the differential equations of their G-invariant forms. The equations of the specific classes associated with supergravity theories give the necessary and sufficient conditions for supersymmetry preservation in the theories[3][4]. If there exist G-invariant forms satisfying the conditions, then they solve the equations of motion of common sectors in type II and heterotic supergravity theory[5][6]. *

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