Physlib.Mathematics.Geometry.Metric.Riemannian.Defs

This definition provides an automatic coercion that allows a Riemannian metric $g$ on a manifold $M$ to be treated as a pseudo-Riemannian metric. Mathematically, this reflects the fact that every Riemannian metric is a specific type of pseudo-Riemannian metric where the signature is positive definite.

Let $g$ be a Riemannian metric on a manifold $M$. For any point $x \in M$ and any non-zero tangent vector $v \in T_x M$, the metric satisfies $g_x(v, v) > 0$.

Let $g$ be a Riemannian metric on a smooth manifold $M$. For any point $x \in M$, the associated quadratic form $Q_x: T_x M \to \mathbb{R}$ on the tangent space at $x$, defined by $Q_x(v) = g_x(v, v)$, is positive definite.

Let $M$ be a manifold equipped with a Riemannian metric $g$. For any point $x \in M$, let $Q_x: T_x M \to \mathbb{R}$ be the quadratic form on the tangent space at $x$ defined by $Q_x(v) = g_x(v, v)$. Then the negative dimension of this quadratic form, denoted $\text{negDim}(Q_x)$, is equal to $0$.

For a Riemannian metric $g$ on a manifold $M$ and a point $x \in M$, this definition provides the `InnerProductSpace.Core` structure for the tangent space $T_x M$. It defines the inner product of two tangent vectors $v, w \in T_x M$ using the metric tensor at $x$, denoted as $g_x(v, w)$. This structure encapsulates the fundamental properties of a real inner product: symmetry ($g_x(v, w) = g_x(w, v)$), linearity in the first argument, and positive definiteness ($g_x(v, v) \geq 0$, with equality if and only if $v = 0$).

Given a manifold $M$ with a Riemannian metric $g$ and a point $x \in M$, this definition endows the tangent space $T_x M$ with the structure of a normed additive commutative group. The norm $\|v\|$ of a tangent vector $v \in T_x M$ is induced by the inner product at $x$ defined by the metric, such that $\|v\| = \sqrt{g_x(v, v)}$.

For a Riemannian metric $g$ on a manifold $M$ and a point $x \in M$, this definition endows the tangent space $T_x M$ with the structure of a real inner product space. The inner product of vectors $v, w \in T_x M$ is defined by the metric tensor at $x$, such that $\langle v, w \rangle = g_x(v, w)$. This structure is built upon the normed additive commutative group structure where the norm is given by $\|v\| = \sqrt{g_x(v, v)}$.

Given a manifold $M$ equipped with a Riemannian metric $g$ and a point $x \in M$, this definition endows the tangent space $T_x M$ with the structure of a real inner product space. The inner product of two tangent vectors $v, w \in T_x M$ is defined by the metric tensor at $x$, denoted $\langle v, w \rangle = g_x(v, w)$. This structure is compatible with the norm $\|v\| = \sqrt{g_x(v, v)}$ and the associated normed additive commutative group structure on $T_x M$.

Given a Riemannian metric $g$ on a manifold $M$, a point $x \in M$, and a tangent vector $v$ in the tangent space $T_x M$, this function returns the norm of $v$, denoted as $\|v\|_g$. It is defined as the square root of the inner product of the vector with itself: $\|v\|_g = \sqrt{g_x(v, v)}$.

Given a manifold $M$ equipped with a Riemannian metric $g$, a point $x \in M$, and a tangent vector $v$ in the tangent space $T_x M$, this function returns the norm $\|v\|$ of the vector. This norm is induced by the inner product $g_x$ at the point $x$, defined such that $\|v\| = \sqrt{g_x(v, v)}$.

Let $M$ be a smooth manifold equipped with a Riemannian metric $g$. For any point $x \in M$ and any tangent vector $v \in T_x M$, the Riemannian norm $\|v\|_g$ (defined as $\sqrt{g_x(v, v)}$) is equal to the norm $\|v\|$ of $v$ as defined by the normed additive commutative group structure on $T_x M$ induced by $g$.

Given a Riemannian metric $g$ on a manifold $M$ and a curve $\gamma: \mathbb{R} \to M$, this function calculates the arc length of the curve between the parameters $t_0$ and $t_1$. It is defined as the integral of the norm of the tangent vector (velocity) along the curve: $$L = \int_{t_0}^{t_1} \left\| \dot{\gamma}(t) \right\|_{g} \, dt$$ where $\dot{\gamma}(t)$ denotes the tangent vector to the curve at parameter $t$, and the norm $\|\cdot\|_g$ is the one induced by the Riemannian metric $g$ at the point $\gamma(t)$, i.e., $\|v\|_g = \sqrt{g_{\gamma(t)}(v, v)}$.