Unveiling The Identity Of Fundamental Tensors In Riemannian Submersions

Riemannian submersions, a cornerstone of differential geometry, provide a powerful framework for understanding the relationships between manifolds of different dimensions. They serve as a bridge, allowing us to project the geometry of a higher-dimensional manifold onto a lower-dimensional one while preserving certain crucial geometric properties. This article delves into the intricacies of Riemannian submersions, focusing specifically on the fundamental tensors, $A$ and $T$, which encapsulate the essence of the submersion's behavior. Understanding these tensors is key to unraveling the geometric structures that arise in various contexts, from theoretical physics to applied mathematics. This exploration aims to clarify the role of these tensors and their interplay within the broader landscape of Riemannian geometry. We will dissect their definitions, properties, and significance, paving the way for a deeper appreciation of Riemannian submersions and their applications.

Riemannian Submersions: A Geometric Projection

At its heart, a Riemannian submersion is a smooth map between Riemannian manifolds that behaves like a projection, preserving the lengths of horizontal vectors. Formally, let $\pi: (M, g) \rightarrow (B, h)$ be a smooth surjective map between Riemannian manifolds $(M, g)$ and $(B, h)$. The map $\pi$ is a Riemannian submersion if it satisfies the following two essential conditions:

1. The differential $\pi_*$ is surjective at each point $p \in M$: This condition ensures that the map projects the tangent space $T_pM$ onto the tangent space $T_{\pi(p)}B$. In simpler terms, it means that the map covers the base manifold $B$ in a smooth and continuous manner.

2. The differential $\pi_*$ preserves the lengths of horizontal vectors: To understand this condition, we first need to define the concept of horizontal and vertical vectors. The vertical space $V_p$ at a point $p \in M$ is the kernel of the differential $\pi_*$, i.e., $V_p = \text{ker}(\pi_*)$. The horizontal space $H_p$ is the orthogonal complement of $V_p$ in $T_pM$ with respect to the Riemannian metric $g$. The condition then states that for any horizontal vector $X \in H_p$, we have $g(X, X) = h(\pi_*(X), \pi_*(X))$. This property is crucial as it ensures that the submersion preserves the local Riemannian structure when projecting horizontal vectors from the total space $M$ to the base space $B$.

Riemannian submersions are a generalization of Riemannian coverings, providing a flexible tool for studying manifolds with related geometries. They find applications in diverse areas, such as Kaluza-Klein theory in physics, where they model the dimensional reduction of spacetime, and in the study of homogeneous spaces in differential geometry. The geometry of a Riemannian submersion is largely determined by the behavior of the horizontal and vertical distributions, and the fundamental tensors $T$ and $A$ serve as the key descriptors of this behavior. These tensors quantify the integrability of the distributions and the curvature properties of the submersion, allowing us to dissect and understand the intricate geometric relationships between the total space and the base space.

The Fundamental Tensors: Unveiling the Submersion's Geometry

The fundamental tensors of a Riemannian submersion, denoted by $T$ and $A$, are bilinear forms that capture the essence of how the horizontal and vertical distributions interact. These tensors, introduced by O'Neill and Gray, provide crucial insights into the geometry of the submersion, revealing how the curvature and integrability properties are affected by the projection. Understanding these tensors is essential for analyzing the geometric structure of the submersion and its impact on the manifolds involved. The tensor $T$ is associated with the integrability of the vertical distribution, while the tensor $A$ describes the integrability of the horizontal distribution and the twisting of the submersion. Together, they provide a comprehensive picture of the submersion's geometric behavior.

Defining the Tensors: A Formal Approach

To formally define the fundamental tensors, let $\pi: (M, g) \rightarrow (B, h)$ be a Riemannian submersion. Let $V$ and $H$ denote the vertical and horizontal distributions, respectively. Given vector fields $E, F$ on $M$, we can decompose their covariant derivatives using the Levi-Civita connection $\nabla$ of $g$ into vertical and horizontal components. The fundamental tensors $T$ and $A$ are then defined as follows:

• The tensor $T$: This tensor measures the obstruction to the integrability of the vertical distribution $V$. For vector fields $E, F$ on $M$, the tensor $T$ is defined as

  $T_E F = \mathcal{H}(\nabla_{\mathcal{V}E} \mathcal{V}F) + \mathcal{V}(\nabla_{\mathcal{V}E} \mathcal{H}F)$,

  where $\mathcal{V}$ and $\mathcal{H}$ denote the vertical and horizontal projections, respectively. The tensor $T$ can be further decomposed into two parts: $T_E F = T^1_E F + T^2_E F$, where $T^1_E F = \mathcal{H}(\nabla_{\mathcal{V}E} \mathcal{V}F)$ and $T^2_E F = \mathcal{V}(\nabla_{\mathcal{V}E} \mathcal{H}F)$. The part $T^1$ measures the obstruction to the integrability of the vertical distribution, while $T^2$ measures the twisting of the horizontal distribution along the vertical distribution.

• The tensor $A$: This tensor is associated with the integrability of the horizontal distribution $H$ and the twisting of the submersion. For vector fields $E, F$ on $M$, the tensor $A$ is defined as

  $A_E F = \mathcal{H}(\nabla_{\mathcal{H}E} \mathcal{V}F) + \mathcal{V}(\nabla_{\mathcal{H}E} \mathcal{H}F)$.

  Similar to $T$, the tensor $A$ can be decomposed into two parts: $A_E F = A^1_E F + A^2_E F$, where $A^1_E F = \mathcal{H}(\nabla_{\mathcal{H}E} \mathcal{V}F)$ and $A^2_E F = \mathcal{V}(\nabla_{\mathcal{H}E} \mathcal{H}F)$. The part $A^1$ measures the twisting of the vertical distribution along the horizontal distribution, while $A^2$ measures the obstruction to the integrability of the horizontal distribution.

These definitions, while seemingly complex, provide a powerful tool for analyzing the geometry of Riemannian submersions. The tensors $T$ and $A$ capture the intricate interplay between the horizontal and vertical distributions, revealing crucial information about the submersion's structure. By examining these tensors, we can gain a deeper understanding of how the geometry of the total space $M$ is projected onto the base space $B$.

Properties and Significance: Decoding the Tensors

The fundamental tensors $T$ and $A$ possess several important properties that shed light on their geometric significance. These properties are essential for understanding the role of the tensors in characterizing Riemannian submersions and their behavior. By analyzing these properties, we can unlock the deeper meaning behind these tensors and their impact on the geometry of the submersion.

• Skew-symmetry: The tensor $A$ exhibits a crucial skew-symmetry property. Specifically, for horizontal vector fields $X$ and $Y$, we have $A_X Y = -A_Y X$. This property implies that $A$ acts as a measure of the non-integrability of the horizontal distribution. If $A$ vanishes identically on horizontal vector fields, then the horizontal distribution is integrable, meaning that there exist horizontal submanifolds in $M$. This skew-symmetry is fundamental to understanding how the horizontal distribution twists and interacts within the submersion.

• Vertical Vector Fields: When applied to vertical vector fields, the tensor $A$ vanishes. This property follows directly from the definition of $A$ and the orthogonality of the horizontal and vertical distributions. It highlights the fact that $A$ primarily captures the interaction between horizontal and vertical vectors, rather than the behavior within the vertical distribution itself. This distinction is crucial for understanding the specific role of $A$ in measuring the twisting of the horizontal distribution.

• Integrability: The tensor $T$ plays a key role in determining the integrability of the vertical distribution. Specifically, the vertical distribution is integrable if and only if the horizontal component of $T$ vanishes, i.e., $T^1 = 0$. This property connects the fundamental tensor $T$ directly to the fundamental geometric property of integrability, allowing us to use $T$ as a tool for analyzing the structure of the submersion. When the vertical distribution is integrable, the fibers of the submersion form a foliation of $M$, which simplifies the analysis of the submersion's geometry.

• Geometric Interpretation: The tensors $T$ and $A$ have clear geometric interpretations. The tensor $A$ can be viewed as measuring the obstruction to the horizontal distribution being a Riemannian foliation. The tensor $T$, on the other hand, measures the second fundamental form of the fibers, which are the submanifolds of $M$ obtained by taking the preimage of points in $B$ under $\pi$. These interpretations provide a visual and intuitive understanding of the tensors, making them more accessible and easier to apply in geometric analysis.

These properties and interpretations provide a deeper understanding of the role of the fundamental tensors in characterizing Riemannian submersions. They allow us to connect the abstract definitions of the tensors to concrete geometric properties, making them a powerful tool for analyzing the structure and behavior of these submersions. By studying the properties of $T$ and $A$, we can unlock valuable insights into the geometric relationships between the total space and the base space of a Riemannian submersion.

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