diff --git a/blackjax/mcmc/metrics.py b/blackjax/mcmc/metrics.py
index 1368a8441..e41ff1d16 100644
--- a/blackjax/mcmc/metrics.py
+++ b/blackjax/mcmc/metrics.py
@@ -28,7 +28,8 @@
 We can also generate a relativistic dynamic :cite:p:`lu2017relativistic`.
 
 """
-from typing import Callable, NamedTuple, Optional, Protocol, Union
+
+from typing import Any, Callable, NamedTuple, Optional, Protocol, Union
 
 import jax.numpy as jnp
 import jax.scipy as jscipy
@@ -38,14 +39,18 @@
 from blackjax.types import Array, ArrayLikeTree, ArrayTree, PRNGKey
 from blackjax.util import generate_gaussian_noise
 
-__all__ = ["default_metric", "gaussian_euclidean", "gaussian_riemannian"]
+__all__ = [
+    "default_metric",
+    "gaussian_euclidean",
+    "gaussian_riemannian",
+    "gaussian_euclidean_low_rank",
+]
 
 
 class KineticEnergy(Protocol):
     def __call__(
         self, momentum: ArrayLikeTree, position: Optional[ArrayLikeTree] = None
-    ) -> float:
-        ...
+    ) -> float: ...
 
 
 class CheckTurning(Protocol):
@@ -56,14 +61,14 @@ def __call__(
         momentum_sum: ArrayLikeTree,
         position_left: Optional[ArrayLikeTree] = None,
         position_right: Optional[ArrayLikeTree] = None,
-    ) -> bool:
-        ...
+    ) -> bool: ...
 
 
 class Metric(NamedTuple):
     sample_momentum: Callable[[PRNGKey, ArrayLikeTree], ArrayLikeTree]
     kinetic_energy: KineticEnergy
     check_turning: CheckTurning
+    data: Any = None
 
 
 MetricTypes = Union[Metric, Array, Callable[[ArrayLikeTree], Array]]
@@ -208,6 +213,120 @@ def is_turning(
     return Metric(momentum_generator, kinetic_energy, is_turning)
 
 
+def gaussian_euclidean_low_rank(
+    diagonal_scale_std: Array,
+    eigenvectors: Array,
+    eigenvalues: Array,
+) -> Metric:
+    r"""Hamiltonian dynamic on euclidean manifold with normally-distributed momentum
+    :cite:p:`betancourt2013general`.
+
+    The gaussian euclidean metric is a euclidean metric further characterized
+    by setting the conditional probability density :math:`\pi(momentum|position)`
+    to follow a standard gaussian distribution. A Newtonian hamiltonian
+    dynamics is assumed.
+
+    This uses the mass matrix $(D^{-1}(V(\Sigma - I)V^T + I)D^{-1})^{-1}$.
+
+    Parameters
+    ----------
+    diagonal_scale_std
+        The diagonal $D^{-1}$. This should for instance correspond to the standard deviation
+        of the posterior.
+    eigenvectors
+        An arbitrary number of eigenvectors
+    eigenvalues
+        The corresponding eigenvalues
+
+    Returns
+    -------
+    momentum_generator
+        A function that generates a value for the momentum at random.
+    kinetic_energy
+        A function that returns the kinetic energy given the momentum.
+    is_turning
+        A function that determines whether a trajectory is turning back on
+        itself given the values of the momentum along the trajectory.
+
+    """
+    (ndim,) = jnp.shape(diagonal_scale_std)
+    (ndim_, n_eigs) = jnp.shape(eigenvectors)
+    if ndim != ndim_:
+        raise ValueError("Shape mismatch in metric.")
+
+    (n_eigs_,) = jnp.shape(eigenvalues)
+    if n_eigs != n_eigs_:
+        raise ValueError("Shape mismatch in metric.")
+
+    # Compute (V(\Sigma - I)V^T + I)x
+    def inner_matrix_mult(vals, vecs, x):
+        projected = x @ vecs
+        scaled = (vals - 1) * projected
+        projected_back = vecs @ scaled
+        return projected_back + x
+
+    def inv_mass_matrix_mult(x):
+        scaled = x * diagonal_scale_std
+        product = inner_matrix_mult(eigenvalues, eigenvectors, scaled)
+        return product * diagonal_scale_std
+
+    def momentum_generator(rng_key: PRNGKey, position: ArrayLikeTree) -> ArrayTree:
+        unit_draws = generate_gaussian_noise(rng_key, position)
+        sqrt_vals = jnp.sqrt(jnp.reciprocal(eigenvalues))
+        sqrt_inv_diag = jnp.sqrt(jnp.reciprocal(diagonal_scale_std))
+        return inner_matrix_mult(sqrt_vals, eigenvectors, unit_draws) * sqrt_inv_diag
+
+    def kinetic_energy(
+        momentum: ArrayLikeTree, position: Optional[ArrayLikeTree] = None
+    ) -> float:
+        del position
+        momentum, _ = ravel_pytree(momentum)
+        velocity = inv_mass_matrix_mult(momentum)
+        kinetic_energy_val = 0.5 * jnp.dot(velocity, momentum)
+        return kinetic_energy_val
+
+    def is_turning(
+        momentum_left: ArrayLikeTree,
+        momentum_right: ArrayLikeTree,
+        momentum_sum: ArrayLikeTree,
+        position_left: Optional[ArrayLikeTree] = None,
+        position_right: Optional[ArrayLikeTree] = None,
+    ) -> bool:
+        """Generalized U-turn criterion :cite:p:`betancourt2013generalizing,nuts_uturn`.
+
+        Parameters
+        ----------
+        momentum_left
+            Momentum of the leftmost point of the trajectory.
+        momentum_right
+            Momentum of the rightmost point of the trajectory.
+        momentum_sum
+            Sum of the momenta along the trajectory.
+
+        """
+        del position_left, position_right
+
+        m_left, _ = ravel_pytree(momentum_left)
+        m_right, _ = ravel_pytree(momentum_right)
+        m_sum, _ = ravel_pytree(momentum_sum)
+
+        velocity_left = inv_mass_matrix_mult(m_left)
+        velocity_right = inv_mass_matrix_mult(m_right)
+
+        # rho = m_sum
+        rho = m_sum - (m_right + m_left) / 2
+        turning_at_left = jnp.dot(velocity_left, rho) <= 0
+        turning_at_right = jnp.dot(velocity_right, rho) <= 0
+        return turning_at_left | turning_at_right
+
+    return Metric(
+        momentum_generator,
+        kinetic_energy,
+        is_turning,
+        data=(diagonal_scale_std, eigenvalues, eigenvectors),
+    )
+
+
 def gaussian_riemannian(
     mass_matrix_fn: Callable,
 ) -> Metric: