Riemannian Submersions Stiefel To Grassmann Manifolds With Geodesic Fibers

Introduction to Riemannian Submersions and Manifolds

In the realm of Riemannian geometry, understanding the relationships between different manifolds is crucial. Among the most insightful tools for exploring these connections is the concept of a Riemannian submersion. A Riemannian submersion is a smooth map between Riemannian manifolds that preserves the lengths of horizontal vectors. These submersions provide a powerful framework for studying the geometry and topology of manifolds by relating them to simpler, well-understood spaces. Specifically, the submersion from the Stiefel manifold to the Grassmann manifold, with totally geodesic fibers, offers a rich area of investigation.

At the heart of this discussion are the Stiefel and Grassmann manifolds, both of which play significant roles in linear algebra, topology, and various applications in engineering and physics. The Stiefel manifold, denoted as $V_k(\mathbb{C}^n)$, comprises orthonormal $k$-frames in $\mathbb{C}^n$, while the Grassmann manifold, denoted as $Gr(k, \mathbb{C}^n)$, consists of $k$-dimensional subspaces in $\mathbb{C}^n$. The natural projection from the Stiefel manifold to the Grassmann manifold, which maps an orthonormal $k$-frame to the $k$-dimensional subspace it spans, is a fundamental example of a submersion. Understanding the geometric properties of this submersion, particularly the nature of its fibers, provides deep insights into the structure of these manifolds.

The significance of studying Riemannian submersions with totally geodesic fibers lies in the fact that these fibers represent the geometric building blocks of the submersion. A submanifold is said to be totally geodesic if every geodesic in the submanifold is also a geodesic in the ambient manifold. In the context of the submersion from the Stiefel manifold to the Grassmann manifold, the fibers are the preimages of points in the Grassmann manifold under the projection map. When these fibers are totally geodesic, the submersion exhibits a particularly nice geometric structure, allowing for a more detailed analysis of the relationship between the Stiefel and Grassmann manifolds.

The unitary matrices, denoted as $U(n)$, with their bi-invariant metric $g(X, Y) = \text{Tr}(Y^\dagger X)$, serve as the backdrop for our exploration. This metric endows $U(n)$ with a rich geometric structure, making it a fertile ground for studying submersions and related geometric concepts. The bi-invariant property of the metric, meaning it is invariant under both left and right multiplication by unitary matrices, ensures that the geometric properties of $U(n)$ are well-behaved and amenable to analysis. This sets the stage for a deeper dive into the submersion from the Stiefel manifold to the Grassmann manifold, revealing the intricate connections between these geometric objects.

Stiefel Manifolds: Orthonormal Frames in Complex Space

The Stiefel manifold, denoted as $V_k(\mathbb{C}^n)$, is a fundamental concept in linear algebra and differential geometry. It is defined as the set of all orthonormal $k$-frames in the complex vector space $\mathbb{C}^n$. An orthonormal $k$-frame is a set of $k$ orthonormal vectors in $\mathbb{C}^n$. Mathematically, we can represent $V_k(\mathbb{C}^n)$ as the set of $n \times k$ matrices $A$ such that $A^\dagger A = I_k$, where $A^\dagger$ denotes the conjugate transpose of $A$ and $I_k$ is the $k \times k$ identity matrix. This condition ensures that the columns of $A$ are orthonormal, forming a $k$-frame in $\mathbb{C}^n$.

Understanding the structure of the Stiefel manifold requires delving into its topological and differential geometric properties. The Stiefel manifold can be viewed as a submanifold of the space of all $n \times k$ matrices, inheriting its smooth structure from the ambient space. This allows us to apply the tools of differential geometry to study its curvature, geodesics, and other geometric invariants. The topology of the Stiefel manifold is also of significant interest, as it provides insights into the connectivity and homotopy groups of the manifold. These topological properties are closely related to the algebraic structure of the Stiefel manifold and its role in various mathematical constructions.

The significance of the Stiefel manifold extends beyond pure mathematics. It finds applications in a variety of fields, including signal processing, computer vision, and machine learning. In signal processing, Stiefel manifolds are used to represent and manipulate sets of orthonormal signals, which are essential for tasks such as beamforming and channel estimation. In computer vision, they arise in the context of pose estimation and object recognition, where the orientation of an object can be represented as a point on a Stiefel manifold. In machine learning, Stiefel manifolds are used in dimensionality reduction techniques and in the training of neural networks with orthogonal weight matrices.

Exploring the properties of Stiefel manifolds often involves considering their relationship with other geometric objects, such as the unitary group $U(n)$ and the Grassmann manifold $Gr(k, \mathbb{C}^n)$. The unitary group, consisting of all $n \times n$ unitary matrices, acts transitively on the Stiefel manifold, meaning that any orthonormal $k$-frame can be obtained from any other by a unitary transformation. This action provides a powerful tool for studying the symmetries of the Stiefel manifold and for constructing invariant geometric quantities. The connection between the Stiefel manifold and the Grassmann manifold, which we will discuss in more detail later, is particularly important. The Grassmann manifold can be viewed as the quotient of the Stiefel manifold by the action of the unitary group $U(k)$, leading to a rich interplay between the geometric structures of these two manifolds.

Grassmann Manifolds: Subspaces of Complex Space

The Grassmann manifold, denoted as $Gr(k, \mathbb{C}^n)$, is the space of all $k$-dimensional subspaces in the complex vector space $\mathbb{C}^n$. This manifold is a cornerstone of linear algebra, differential geometry, and topology, offering a rich geometric structure that encodes fundamental properties of vector spaces and their subspaces. Understanding the Grassmann manifold is essential for a wide range of applications, from pure mathematics to engineering and physics.

To visualize the Grassmann manifold, consider the set of all $k$-dimensional subspaces within $\mathbb{C}^n$. Each point on the Grassmann manifold represents one such subspace. For example, $Gr(1, \mathbb{C}^n)$ is the space of all lines through the origin in $\mathbb{C}^n$, which is the complex projective space $\mathbb{CP}^{n-1}$. Similarly, $Gr(2, \mathbb{C}^4)$ represents the space of all two-dimensional planes in $\mathbb{C}^4$. The Grassmann manifold is a smooth manifold, and its dimension can be calculated as $k(n-k)$. This dimension reflects the degrees of freedom required to specify a $k$-dimensional subspace within an $n$-dimensional space.

The construction of the Grassmann manifold involves several equivalent approaches, each providing a unique perspective on its structure. One common approach is to view $Gr(k, \mathbb{C}^n)$ as a quotient space of the Stiefel manifold $V_k(\mathbb{C}^n)$. Specifically, we can consider the action of the unitary group $U(k)$ on $V_k(\mathbb{C}^n)$ by right multiplication. This action corresponds to changing the orthonormal basis of a $k$-dimensional subspace, while leaving the subspace itself unchanged. The Grassmann manifold is then the quotient space obtained by identifying points in $V_k(\mathbb{C}^n)$ that differ by the action of $U(k)$. This quotient construction endows the Grassmann manifold with a natural smooth structure and a Riemannian metric inherited from the Stiefel manifold.

Applications of Grassmann manifolds are widespread across various scientific disciplines. In linear algebra, they provide a geometric framework for studying subspaces and their relationships. In differential geometry, they serve as model spaces for studying curvature and other geometric invariants. In topology, they play a crucial role in the classification of vector bundles and the study of characteristic classes. Beyond mathematics, Grassmann manifolds find applications in signal processing, computer vision, and machine learning. For example, they are used in dimensionality reduction techniques, where high-dimensional data is projected onto lower-dimensional subspaces, and in the analysis of image and video data, where subspaces can represent patterns and features in the data.

The Bi-Invariant Metric on Unitary Matrices

Unitary matrices, denoted as $U(n)$, form a crucial group in mathematics and physics, particularly in quantum mechanics and representation theory. Understanding the geometry of the unitary group requires endowing it with a suitable metric. A bi-invariant metric, denoted as $g(X, Y) = \text{Tr}(Y^\dagger X)$, provides an elegant and powerful way to achieve this. This metric is bi-invariant because it remains unchanged under both left and right multiplication by unitary matrices, a property that simplifies many calculations and provides deep insights into the group's structure.

The bi-invariant metric on the unitary group is defined using the trace inner product. For any two tangent vectors $X$ and $Y$ at a point $U$ in $U(n)$, the metric $g(X, Y)$ is given by the trace of the product of the adjoint of $Y$ and $X$. Mathematically, this is expressed as $g(X, Y) = \text{Tr}(Y^\dagger X)$. Here, $X$ and $Y$ are elements of the Lie algebra $\mathfrak{u}(n)$, which consists of skew-Hermitian matrices. The trace inner product has several desirable properties, including being positive definite, symmetric, and invariant under unitary transformations. These properties make it an ideal choice for defining a Riemannian metric on $U(n)$.

The significance of bi-invariance cannot be overstated. A bi-invariant metric ensures that the geometric properties of the unitary group are uniform across the entire manifold. This means that the curvature, geodesics, and other geometric invariants are the same at every point in $U(n)$. This uniformity simplifies many calculations and allows for a more intuitive understanding of the group's geometry. For example, the geodesics on $U(n)$ with respect to the bi-invariant metric are simply one-parameter subgroups, which are easy to compute and visualize. Moreover, the bi-invariance property implies that the left and right translations on $U(n)$ are isometries, meaning they preserve distances and angles. This symmetry is fundamental to many applications of unitary matrices in physics and engineering.

The bi-invariant metric on $U(n)$ also plays a crucial role in the study of Riemannian submersions. When considering submersions from the unitary group to other manifolds, the bi-invariant metric provides a natural and well-behaved Riemannian structure on the total space. This simplifies the analysis of the submersion and allows for a deeper understanding of the relationship between the geometries of the total space and the base space. In the context of the submersion from the Stiefel manifold to the Grassmann manifold, the bi-invariant metric on $U(n)$ is essential for demonstrating that the fibers of the submersion are totally geodesic, a property that has significant implications for the geometry of these manifolds.

Riemannian Submersions: Projecting Geometry

Riemannian submersions are powerful tools in differential geometry that allow us to relate the geometries of different manifolds. A Riemannian submersion is a smooth map between Riemannian manifolds that preserves the lengths of horizontal vectors. This concept, introduced by Arthur Besse in his seminal book on Einstein manifolds, provides a way to project the geometry of a higher-dimensional manifold onto a lower-dimensional one, while preserving certain key geometric properties. Understanding Riemannian submersions is crucial for studying the relationships between different geometric spaces and for constructing new examples of manifolds with specific geometric properties.

Formally, a smooth map $\pi: M \rightarrow B$ between Riemannian manifolds $(M, g_M)$ and $(B, g_B)$ is a Riemannian submersion if it satisfies two main conditions. First, the map $\pi$ must be a submersion, meaning that its differential $d\pi_x: T_xM \rightarrow T_{\pi(x)}B$ is surjective for all points $x$ in $M$. This condition ensures that the tangent spaces of $M$ are mapped onto the tangent spaces of $B$. Second, the differential $d\pi_x$ must preserve the lengths of horizontal vectors. To define horizontal vectors, we consider the vertical space $V_x = \text{ker}(d\pi_x)$, which is the subspace of $T_xM$ consisting of vectors that are mapped to zero by $d\pi_x$. The horizontal space $H_x$ is then defined as the orthogonal complement of $V_x$ in $T_xM$ with respect to the Riemannian metric $g_M$. The length-preserving condition means that for any horizontal vector $X \in H_x$, we have $g_M(X, X) = g_B(d\pi_x(X), d\pi_x(X))$.

The fibers of a Riemannian submersion, which are the preimages of points in the base manifold $B$, play a crucial role in understanding the geometry of the submersion. The vertical spaces $V_x$ are precisely the tangent spaces to the fibers. If the fibers are totally geodesic submanifolds, then every geodesic in a fiber is also a geodesic in the total space $M$. This property has significant implications for the curvature and other geometric invariants of $M$ and $B$. In particular, the O'Neill tensor, which measures the integrability of the horizontal distribution, provides a way to relate the curvatures of $M$, $B$, and the fibers.

Examples of Riemannian submersions abound in differential geometry. One classic example is the Hopf fibration, which is a submersion from the 3-sphere $S^3$ to the 2-sphere $S^2$ with fibers that are circles. Another important example is the projection from a principal bundle to its base manifold, where the fibers are the orbits of the group action. In the context of this discussion, the projection from the Stiefel manifold to the Grassmann manifold is a fundamental example of a Riemannian submersion. Understanding the properties of this submersion, particularly the nature of its fibers, provides deep insights into the structure of these manifolds and their relationships.

Submersion from Stiefel to Grassmann: Totally Geodesic Fibers

The submersion from the Stiefel manifold $V_k(\mathbb{C}^n)$ to the Grassmann manifold $Gr(k, \mathbb{C}^n)$ is a cornerstone example in Riemannian geometry. This submersion, denoted by $\pi: V_k(\mathbb{C}^n) \rightarrow Gr(k, \mathbb{C}^n)$, maps an orthonormal $k$-frame in $\mathbb{C}^n$ to the $k$-dimensional subspace it spans. The fibers of this submersion, which are the preimages of points in the Grassmann manifold, have a particularly nice geometric property: they are totally geodesic submanifolds. This property makes the submersion from the Stiefel manifold to the Grassmann manifold a valuable tool for studying the geometry of these spaces and their relationship to each other.

To understand the submersion, consider a point $A$ in the Stiefel manifold $V_k(\mathbb{C}^n)$, which is an $n \times k$ matrix with orthonormal columns. The submersion $\pi$ maps $A$ to the $k$-dimensional subspace of $\mathbb{C}^n$ spanned by the columns of $A$. This subspace is a point in the Grassmann manifold $Gr(k, \mathbb{C}^n)$. The fiber over a point $P$ in $Gr(k, \mathbb{C}^n)$ consists of all orthonormal $k$-frames that span the subspace $P$. This fiber can be identified with the unitary group $U(k)$, since changing the orthonormal basis of $P$ corresponds to multiplying $A$ on the right by a unitary matrix in $U(k)$.

The key result is that these fibers are totally geodesic. A submanifold is totally geodesic if every geodesic in the submanifold is also a geodesic in the ambient manifold. In the context of the submersion from the Stiefel manifold to the Grassmann manifold, this means that if we take a geodesic in a fiber (which is a copy of $U(k)$) and view it as a curve in the Stiefel manifold, it is also a geodesic in the Stiefel manifold. This property has significant implications for the geometry of the submersion. It implies that the fibers are, in a sense, geometrically