On the frame bundle adapted to a submanifold

Abstract

Let M be a submanifold of a Riemannian manifold (N,g). M induces a subbundle O(M,N) of adapted frames over M of the bundle of orthonormal frames O(N). The Riemannian metric g induces a natural metric on O(N). We study the geometry of a submanifold O(M,N) in O(N). We characterize the horizontal distribution of O(M,N) and state its correspondence with the horizontal lift in O(N) induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.

Geometry of G-Structures via the Intrinsic Torsion

“…In this section we will introduce invertible tensor induced by the intrinsic torsion, the Riemannian metric defined by this tensor and state the properties of the Levi-Civita connection of this new metric. Results in this section are generalizations of the results obtained by the author in [18]. Adopt the notation from the previous section.…”

“…In order to describe explicitly the condition for the harmonicity of σ let us introduce certain curvature operator [18]. For any α ∈ so(M ) let…”

“…In this case we need to consider the Levi-Civita connection ∇ SO(M ) of the Riemannian metric g SO(M ) . One can show [18] that…”

“…More generally, we use the concept of intrinsic torsion to obtain some results on the geometry of G-structures. The idea comes from the results obtained by the author [18] in the case of a single submanifold. We deal with the intrinsic and extrinsic geometry of a Gstructure.…”

Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds

“…The integrability condition for the diagonal and horizontal lifts of the metallic structure J on FM is established. The geometric structures on FM have been studied by Cordero et al [37], Kowalski [38], Sekizawa [39], Kowalski and Sekizawa [40], Niedzialomski [41], Lachieze-Rey [42], Khan [43][44][45] and many more.…”