Three-Dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds

מושגי ליבה

A 3-dimensional Riemannian manifold (M, g, Q) with a tensor structure Q whose fourth power is the identity is associated with a Riemannian almost product manifold (M, g, P), where P = Q^2. The almost product manifold (M, g, P) is shown to belong to the class of locally conformal Riemannian product manifolds.

תקציר

The paper studies the geometric properties of a 3-dimensional Riemannian manifold (M, g, Q) equipped with a tensor structure Q whose fourth power is the identity. An associated Riemannian almost product manifold (M, g, P) is also investigated, where P = Q^2.

Key highlights:

• The manifold (M, g, P) is shown to be a locally conformal Riemannian product manifold, with the fundamental tensor F satisfying a specific identity.
• Necessary and sufficient conditions are obtained for the structures Q and P to be parallel with respect to the Levi-Civita connection.
• Classes of almost Einstein and Einstein manifolds are determined, and their curvature properties are studied.
• Special bases in the tangent space TpM are introduced, and relations between Ricci curvatures and sectional curvatures are established.
• An example of the considered manifolds is provided by a 3-dimensional catenoid hypersurface embedded in a 4-dimensional Euclidean space.

סטטיסטיקה

A > 0, B > 0
x3(x1)^2 + (x2)^2 ≠ 0
cos φ > 0, cos ψ < cos φ, where φ = ∠(x, Qx) and ψ = ∠(x, Q^2x)

ציטוטים

"Every basis of type {x, Qx, Q^2x}, {x, Q^2x, Q^3x}, {x, Qx, Q^3x} and {Qx, Q^2x, Q^3x} of TpM (p ∈ M) is called a Q-basis. In this case we say that the vector x induces a Q-basis of TpM."
"If the Ricci tensor ρ on (M, g, Q) has the form (36), then the sectional curvatures of the 2-planes, determined by the basis vectors, are k(x, Qx) = k(Qx, Q^2x) = k(x, Q^3x) = k(Q^2x, Q^3x) = k(x, Q^2x) = k(Qx, Q^3x) = -τ/6."

תובנות מפתח מזוקקות מ:

Three-dimensional Riemannian manifolds associated with locally conformal Riemannian product manifolds

by Iva Dokuzova ב- arxiv.org 10-01-2024

https://arxiv.org/pdf/2409.19626.pdf

שאלות מעמיקות

What are the implications of the locally conformal Riemannian product structure on the global topology and geometry of the manifold (M, g, P)?

The locally conformal Riemannian product structure on the manifold ((M, g, P)) has significant implications for its global topology and geometry. This structure indicates that the manifold can be locally expressed as a product of Riemannian manifolds, where the metric (g) can be transformed into a conformally equivalent metric. The existence of such a structure suggests that the manifold possesses a rich geometric framework, allowing for the application of various geometric techniques and results.

Topology: The locally conformal Riemannian product structure can influence the manifold's global topology by allowing for the existence of non-trivial topological features. For instance, the classification of Riemannian manifolds with locally conformal structures often leads to insights about their fundamental groups and covering spaces. The presence of a parallel structure, as indicated by the conditions derived in the paper, can also imply certain topological constraints, such as the existence of specific types of fibers or bundles over the manifold.

Geometry: The curvature properties of ((M, g, P)) are closely tied to its locally conformal structure. The fundamental tensor (F) and the associated curvature tensors provide a framework for understanding how curvature behaves under conformal transformations. The results obtained in the paper, particularly regarding the Ricci tensor and sectional curvatures, reveal that the geometry of ((M, g, P)) is influenced by the interplay between the metrics (g) and (\tilde{g}). This relationship can lead to insights into the manifold's curvature, such as conditions for being an Einstein or almost Einstein manifold, which are crucial for understanding the manifold's geometric properties.

Applications: The implications of this structure extend to various applications in differential geometry and theoretical physics, particularly in the study of general relativity, where locally conformal structures can model spacetime geometries. The results can also be applied to study the stability of geometric structures under perturbations, providing a deeper understanding of the manifold's behavior under various geometric flows.

How can the curvature properties of the manifold (M, g, Q) be further generalized or extended to higher-dimensional Riemannian manifolds with similar tensor structures?

The curvature properties of the manifold ((M, g, Q)) can be generalized to higher-dimensional Riemannian manifolds by extending the techniques and results derived in the context of three-dimensional manifolds to (n)-dimensional settings. This involves several key steps:

Higher-Dimensional Tensor Structures: In higher dimensions, one can consider tensor structures of type ((1, 1)) that satisfy similar properties, such as (Q^4 = \text{id}). The classification of these structures can be approached using similar methods as those employed in the three-dimensional case, focusing on the compatibility of the tensor (Q) with the Riemannian metric (g).

Curvature Tensor Generalization: The curvature tensor (R) can be expressed in terms of the Ricci tensor and scalar curvature in higher dimensions. The relationships established in the three-dimensional case, such as those involving the Ricci tensor and the associated metrics, can be adapted to higher dimensions by considering the appropriate generalizations of the curvature formulas. This includes extending the definitions of sectional curvature and Ricci curvature to accommodate the additional dimensions.

Locally Conformal Structures: The concept of locally conformal Riemannian product manifolds can be extended to higher dimensions, allowing for the exploration of how these structures interact with the curvature properties of the manifold. The results regarding the fundamental tensor (F) and its implications for curvature can be generalized, leading to new insights into the geometry of higher-dimensional manifolds.

Applications in Geometry and Physics: The generalization of these curvature properties can have implications in various fields, including mathematical physics, where higher-dimensional Riemannian manifolds are often used to model complex systems. The insights gained from studying the curvature properties of ((M, g, Q)) can inform the understanding of more complex geometric structures, such as those encountered in string theory or higher-dimensional general relativity.

Can the techniques and insights developed in this work be applied to study the geometry of other classes of Riemannian manifolds with additional structures, such as almost Hermitian or almost contact manifolds?

Yes, the techniques and insights developed in this work can be effectively applied to study the geometry of other classes of Riemannian manifolds with additional structures, such as almost Hermitian or almost contact manifolds. Here are several ways in which these techniques can be utilized:

Fundamental Tensor Analysis: The analysis of the fundamental tensor (F) and its properties can be adapted to almost Hermitian manifolds, where the additional structure involves a compatible almost complex structure. The relationships between the curvature tensors in the context of locally conformal Riemannian products can provide insights into the curvature properties of almost Hermitian manifolds, particularly in understanding how the complex structure interacts with the Riemannian metric.

Curvature Relations: The curvature relations established for ((M, g, Q)) can be extended to almost contact manifolds, where the geometry is influenced by a contact structure. The techniques used to derive the relationships between the Ricci tensor and the scalar curvature can be similarly applied to study the curvature properties of almost contact manifolds, leading to a deeper understanding of their geometric characteristics.

Classification and Structure Theorems: The classification results obtained for locally conformal Riemannian product manifolds can inspire similar classification efforts for almost Hermitian and almost contact manifolds. By examining the conditions under which certain curvature properties hold, one can derive structure theorems that characterize these manifolds based on their geometric properties.

Applications in Differential Geometry: The insights gained from studying the curvature properties of ((M, g, Q)) can inform broader research in differential geometry, particularly in the context of manifolds with additional structures. The techniques developed in this work can be applied to explore the stability of geometric structures, the existence of special types of geodesics, and the behavior of curvature under various geometric flows.

In summary, the methodologies and findings from the study of ((M, g, Q)) provide a robust framework for exploring the geometry of various classes of Riemannian manifolds with additional structures, enriching the field of differential geometry and its applications.