geomstats · luisfpereira · Dec 23, 2023 · Dec 22, 2023 · Dec 22, 2023 · Dec 22, 2023
diff --git a/geomstats/datasets/utils.py b/geomstats/datasets/utils.py
@@ -424,7 +424,7 @@ def load_mammals(file_path=MAMMALS_PATH):
     ----------
     .. [Franz2015]  Franz, M., Altmann, J., & Alberts, S. C.
         "Knockouts of high-ranking males have limited impact on baboon social networks."
-         Current Zoology, 61(1), 107-113, 2015.
+        Current Zoology, 61(1), 107-113, 2015.
     .. [Rossi2015]  Rossi, R., & Ahmed, N.
         "The network data repository with interactive graph analytics and
         visualization." In Twenty-ninth AAAI conference on artificial intelligence, 2015

diff --git a/geomstats/distributions/lognormal.py b/geomstats/distributions/lognormal.py
@@ -111,16 +111,15 @@ class LogNormal:
     space : Manifold obj, {Euclidean(n), SPDMatrices(n)}
         Manifold to sample over. Manifold should
         be instance of Euclidean or SPDMatrices.
-    mean : array-like,
-            shape=[dim] if space is Euclidean Space
+    mean : array-like, \
+            shape=[dim] if space is Euclidean Space \
             shape=[n, n] if space is SPD Manifold
         Mean of the distribution.
-    cov : array-like,
-            shape=[dim, dim] if space is Euclidean Space
+    cov : array-like, \
+            shape=[dim, dim] if space is Euclidean Space \
             shape=[n*(n+1)/2, n*(n+1)/2] if space is SPD Manifold
         Covariance of the distribution.
 
-
     Example
     --------
     >>> import geomstats.backend as gs

diff --git a/geomstats/geometry/connection.py b/geomstats/geometry/connection.py
@@ -357,30 +357,28 @@ def ladder_parallel_transport(
     def riemann_tensor(self, base_point):
         r"""Compute Riemannian tensor at base_point.
 
-        In the literature the riemannian curvature tensor is noted :math:`R_{ijk}^l`.
+        In the literature the Riemannian curvature tensor is noted :math:`R_{ijk}^l`.
 
         Following tensor index convention (ref. Wikipedia), we have:
         :math:`R_{ijk}^l = dx^l(R(X_j, X_k)X_i)`
 
-        which gives :math:`R_{ijk}^lk` as a sum of four terms:
-        :math:`R_{ijk}^l =
-        :math:`\partial_j \Gamma^l_{ki}`
-        :math:`- \partial_k \Gamma^l_{ji}`
-        :math:`+ \Gamma^l_{jm} \Gamma^m_{ki}`
-        :math:`- \Gamma^l_{km} \Gamma^m_{ji}`
+        which gives :math:`R_{ijk}^l` as a sum of four terms:
+
+        .. math::
+            \partial_j \Gamma^l_{ki} - \partial_k \Gamma^l_{ji}
+            + \Gamma^l_{jm} \Gamma^m_{ki} - \Gamma^l_{km} \Gamma^m_{ji}
 
         Note that geomstats puts the contravariant index on
         the first dimension of the Christoffel symbols.
 
         Parameters
         ----------
-        base_point :  array-like, shape=[..., dim]
+        base_point : array-like, shape=[..., dim]
             Point on the manifold.
 
         Returns
         -------
-        riemann_curvature : array-like, shape=[..., dim, dim,
-                                                    dim, dim]
+        riemann_curvature : array-like, shape=[..., dim, dim, dim, dim]
             riemann_tensor[...,i,j,k,l] = R_{ijk}^l
             Riemannian tensor curvature,
             with the contravariant index on the last dimension.
@@ -408,9 +406,11 @@ def curvature(self, tangent_vec_a, tangent_vec_b, tangent_vec_c, base_point):
         r"""Compute the Riemann curvature map R.
 
         For three tangent vectors at base point :math:`P`:
+
         - :math:`X|_P = tangent\_vec\_a`,
         - :math:`Y|_P = tangent\_vec\_b`,
         - :math:`Z|_P = tangent\_vec\_c`,
+
         the curvature(X, Y, Z, P) is defined by
         :math:`R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_[X, Y]Z`.
 
@@ -471,8 +471,10 @@ def directional_curvature(self, tangent_vec_a, tangent_vec_b, base_point):
         r"""Compute the directional curvature (tidal force operator).
 
         For two tangent vectors at base_point :math:`P`:
+
         - :math:`X|_P = tangent\_vec\_a`,
         - :math:`Y|_P = tangent\_vec\_b`,
+
         the directional curvature, better known
         in relativity as the tidal force operator,
         is defined by
@@ -505,10 +507,12 @@ def curvature_derivative(
         r"""Compute the covariant derivative of the curvature.
 
         For four tangent vectors at base_point :math:`P`:
+
         - :math:`H|_P = tangent\_vec\_a`,
         - :math:`X|_P = tangent\_vec\_b`,
         - :math:`Y|_P = tangent\_vec\_c`,
         - :math:`Z|_P = tangent\_vec\_d`,
+
         the covariant derivative of the curvature is defined as:
         :math:`(\nabla_H R)(X, Y) Z |_P`.
 
@@ -540,8 +544,10 @@ def directional_curvature_derivative(
         r"""Compute the covariant derivative of the directional curvature.
 
         For tangent vector fields at base_point :math:`P`:
+
         - :math:`X|_P = tangent\_vec\_a`,
         - :math:`Y|_P = tangent\_vec\_b`,
+
         the covariant derivative (in the direction `X`)
         :math:`(\nabla_X R_Y)(X) |_P = (\nabla_X R)(Y, X) Y |_P` of the
         directional curvature (in the direction `Y`)

diff --git a/geomstats/geometry/diffeo.py b/geomstats/geometry/diffeo.py
@@ -12,7 +12,7 @@ class Diffeo:
 
     Let :math:`f` be the diffeomorphism
     :math:`f: M \rightarrow N` of the manifold
-    :math:`M` into the manifold `N`.
+    :math:`M` into the manifold :math:`N`.
     """
 
     @abc.abstractmethod
@@ -34,7 +34,7 @@ def diffeomorphism(self, base_point):
     def inverse_diffeomorphism(self, image_point):
         r"""Inverse diffeomorphism at base point.
 
-        :math:`f^-1: N \rightarrow M`
+        :math:`f^{-1}: N \rightarrow M`
 
         Parameters
         ----------

diff --git a/geomstats/geometry/discrete_curves.py b/geomstats/geometry/discrete_curves.py
@@ -344,6 +344,7 @@ def inverse_diffeomorphism(self, image_point):
             c(t) = c(0) + \int_0^t q(s) |q(s)|ds
 
         with:
+
         - c the curve that can be retrieved only up to a translation,
         - q the srv representation of the curve,
         - c(0) the starting point of the curve.
@@ -376,7 +377,7 @@ def inverse_diffeomorphism(self, image_point):
     def tangent_diffeomorphism(self, tangent_vec, base_point=None, image_point=None):
         r"""Differential of the square root velocity transform.
 
-        ..math::
+        .. math::
             (h, c) -> dQ_c(h) = |c'|^(-1/2} * (h' - 1/2 * <h',v>v)
             v = c'/|c'|
 
@@ -431,7 +432,7 @@ def inverse_tangent_diffeomorphism(
         r"""Inverse of differential of the square root velocity transform.
 
         .. math::
-            (c, k) -> h, where dQ_c(h)=k and h' = |c'| * (k + <k,v> v)
+            (c, k) -> h, \text{ where } dQ_c(h)=k \text{ and } h' = |c'| * (k + <k,v> v)
 
         Parameters
         ----------
@@ -521,9 +522,11 @@ class FTransform(AutodiffDiffeo):
     f_transform is a bijection if and only if a/2b=1.
 
     If a/2b is an integer not equal to 1:
+
     - then f_transform is well-defined but many-to-one.
 
     If a/2b is not an integer:
+
     - then f_transform is multivalued,
     - and f_transform takes finitely many values if and only if a 2b is rational.
     """
@@ -702,9 +705,11 @@ def riemann_sum(func):
         Compute the left Riemann sum approximation of the integral of a
         function func defined on the unit interval, given by sample points
         at regularly spaced times
-        ..math::
+
+        .. math::
             t_i = i / (k_landmarks),
             i = 0, ..., k_landmarks - 1
+
         (last time is missing).
 
         Parameters
@@ -801,8 +806,10 @@ class SRVTranslationMetric(PullbackDiffeoMetric):
     """Square Root Velocity metric on the space of discrete curves.
 
     The SRV metric is equivalent to the elastic metric chosen with
+
     - bending parameter a = 1,
     - stretching parameter b = 1/2.
+
     It can be obtained as the pullback of the L2 metric by the Square Root
     Velocity Function.
 
@@ -1168,11 +1175,12 @@ class DynamicProgrammingAligner:
     the L2 scalar product between initial_srv and end_srv@gamma where initial_srv
     is the SRV representation of the initial curve and end_srv@gamma is the SRV
     representation of the end curve reparametrized by gamma, i.e
+
     .. math::
-    end_srv@\gamma(t) = end_srv(\gamma(t))\cdot|\gamma(t)|^\frac{1}{2}
+        end_srv@\gamma(t) = end_srv(\gamma(t))\cdot|\gamma(t)|^\frac{1}{2}
 
     The dynamic programming algorithm assumes that for every subinterval
-    :math: '\left[\frac{i}{n},\frac{i+1}{n}\right]' of :math: '\left[0,1\right]',
+    :math:`\left[\frac{i}{n},\frac{i+1}{n}\right]` of :math:`\left[0,1\right]`,
     gamma is linear.
 
     Parameters
@@ -1221,10 +1229,12 @@ def _compute_integral_restricted(srv_1, srv_2, x_min, x_max, y_min, y_max):
         r"""Compute the value of an integral over a subinterval.
 
         Compute n * the value of the integral of
+
         .. math::
         srv_1(t)\cdotsrv_2(\gamma(t))\cdot|\gamma(t)|^\frac{1}{2}
-        over :math: '\left[\x_min,x_max\right]' where gamma restricted to
-        :math: '\left[\x_min,x_max\right]' is a linear.
+
+        over :math:`\left[\x_min,x_max\right]` where gamma restricted to
+        :math:`\left[\x_min,x_max\right]` is a linear.
 
         Parameters
         ----------

diff --git a/geomstats/geometry/discrete_surfaces.py b/geomstats/geometry/discrete_surfaces.py
@@ -14,14 +14,15 @@
 class DiscreteSurfaces(Manifold):
     r"""Space of parameterized discrete surfaces.
 
-    Each surface is sampled with fixed n_vertices vertices and n_faces faces
-    in $\mathbb{R}^3$.
+    Each surface is sampled with fixed `n_vertices` vertices and `n_faces`
+    faces in :math:`\mathbb{R}^3`.
 
-    Each individual surface is represented by a 2d-array of shape `[
-    n_vertices, 3]`. This space corresponds to the space of immersions
+    Each individual surface is represented by a 2d-array of shape
+    `[n_vertices, 3]`. This space corresponds to the space of immersions
     defined below, i.e. the
-    space of smooth functions from a template to manifold $M$ into  $\mathbb{R}^3$,
-    with non-vanishing Jacobian.
+    space of smooth functions from a template to manifold :math:`M`
+    into :math:`\mathbb{R}^3`, with non-vanishing Jacobian.
+
     .. math::
         Imm(M,\mathbb{R}^3)=\{ f \in C^{\infty}(M, \mathbb{R}^3)
         \|Df(x)\|\neq 0 \forall x \in M \}.
@@ -366,8 +367,8 @@ def laplacian(self, point):
         r"""Compute the mesh Laplacian operator of a triangulated surface.
 
         Denoting q the surface, i.e. the point in the DiscreteSurfaces manifold,
-        the laplacian at q is defined as the operator:
-        :math: `\Delta_q = - Tr(g_q^{-1} \nabla^2)`
+        the laplacian at :math:`q` is defined as the operator:
+        :math:`\Delta_q = - Tr(g_q^{-1} \nabla^2)`
         where :math:`g_q` is the surface metric matrix of :math:`q`.
 
         Parameters
@@ -467,8 +468,8 @@ def _laplacian(tangent_vec):
 class ElasticMetric(RiemannianMetric):
     """Elastic metric defined by a family of second order Sobolev metrics.
 
-    Each individual discrete surface is represented by a 2D-array of shape `[
-    n_vertices, 3]`. See [HSKCB2022]_ for details.
+    Each individual discrete surface is represented by a 2D-array of shape
+    `[n_vertices, 3]`. See [HSKCB2022]_ for details.
 
     The parameters a0, a1, b1, c1, d1, a2 (detailed below) are non-negative weighting
     coefficients for the different terms in the metric.
@@ -525,7 +526,7 @@ def _inner_product_a0(self, tangent_vec_a, tangent_vec_b, vertex_areas_bp):
         :math:`\int_M (G_{a_0} + G_{a_1} + G_{b_1} + G_{c_1} + G_{d_1} + G_{a_2})vol_q`.
 
         This method computes :math:`G_{a_0} = a_0 <h, k>`,
-        with notations taken from .. [HSKCB2022].
+        with notations taken from [HSKCB2022]_.
 
         Parameters
         ----------
@@ -563,7 +564,7 @@ def _inner_product_a1(self, ginvdga, ginvdgb, areas_bp):
         :math:`\int_M (G_{a_0} + G_{a_1} + G_{b_1} + G_{c_1} + G_{d_1} + G_{a_2})vol_q`.
 
         This method computes :math:`G_{a_1} = a_1.g_q^{-1} <dh_m, dk_m>`,
-        with notations taken from .. [HSKCB2022].
+        with notations taken from [HSKCB2022]_.
 
         Parameters
         ----------
@@ -602,7 +603,7 @@ def _inner_product_b1(self, ginvdga, ginvdgb, areas_bp):
         :math:`\int_M (G_{a_0} + G_{a_1} + G_{b_1} + G_{c_1} + G_{d_1} + G_{a_2})vol_q`.
 
         This method computes :math:`G_{b_1} = b_1.g_q^{-1} <dh_+, dk_+>`,
-        with notations taken from .. [HSKCB2022].
+        with notations taken from [HSKCB2022]_.
 
         Parameters
         ----------
@@ -643,7 +644,7 @@ def _inner_product_c1(self, point_a, point_b, normals_bp, areas_bp):
         :math:`\int_M (G_{a_0} + G_{a_1} + G_{b_1} + G_{c_1} + G_{d_1} + G_{a_2})vol_q`.
 
         This method computes :math:`G_{c_1} = c_1.g_q^{-1} <dh_\perp, dk_\perp>`,
-        with notations taken from .. [HSKCB2022].
+        with notations taken from [HSKCB2022]_.
 
         Parameters
         ----------
@@ -685,7 +686,7 @@ def _inner_product_d1(
         :math:`\int_M (G_{a_0} + G_{a_1} + G_{b_1} + G_{c_1} + G_{d_1} + G_{a_2})vol_q`.
 
         This method computes :math:`G_{d_1} = d_1.g_q^{-1} <dh_0, dk_0>`,
-        with notations taken from .. [HSKCB2022].
+        with notations taken from [HSKCB2022]_.
 
         Parameters
         ----------
@@ -755,7 +756,7 @@ def _inner_product_a2(
         :math:`\int_M (G_{a_0} + G_{a_1} + G_{b_1} + G_{c_1} + G_{d_1} + G_{a_2})vol_q`.
 
         This method computes :math:`G_{a_2} = a_2 <\Delta_q h, \Delta_q k>`,
-        with notations taken from .. [HSKCB2022].
+        with notations taken from [HSKCB2022]_.
 
         Parameters
         ----------
@@ -803,14 +804,15 @@ def inner_product(self, tangent_vec_a, tangent_vec_b, base_point):
         :math:`\int_M (G_{a_0} + G_{a_1} + G_{b_1} + G_{c_1} + G_{d_1} + G_{a_2})vol_q`
 
         where:
+
         - :math:`G_{a_0} = a_0 <h, k>`
         - :math:`G_{a_1} = a_1.g_q^{-1} <dh_m, dk_m>`
         - :math:`G_{b_1} = b_1.g_q^{-1} <dh_+, dk_+>`
         - :math:`G_{c_1} = c_1.g_q^{-1} <dh_\perp, dk_\perp>`
         - :math:`G_{d_1} = d_1.g_q^{-1} <dh_0, dk_0>`
         - :math:`G_{a_2} = a_2 <\Delta_q h, \Delta_q k>`
 
-        with notations taken from .. [HSKCB2022].
+        with notations taken from [HSKCB2022]_.
 
         Parameters
         ----------

diff --git a/geomstats/geometry/fiber_bundle.py b/geomstats/geometry/fiber_bundle.py
@@ -344,7 +344,7 @@ def integrability_tensor(self, tangent_vec_a, tangent_vec_b, base_point):
 
         The fundamental integrability tensor A is defined for tangent vectors
         :math:`X = tangent\_vec\_a` and :math:`Y = tangent\_vec\_b` of the
-        total space by [O'Neill]_ as
+        total space by [ONeill]_ as
         :math:`A_X Y = ver\nabla_{hor X} (hor Y) + hor \nabla_{hor X}( ver Y)`
         where :math:`hor, ver` are the horizontal and vertical projections
         and :math:`\nabla` is the connection of the total space.
@@ -366,7 +366,7 @@ def integrability_tensor(self, tangent_vec_a, tangent_vec_b, base_point):
 
         References
         ----------
-        .. [O'Neill]  O’Neill, Barrett. The Fundamental Equations of a
+        .. [ONeill]  O’Neill, Barrett. The Fundamental Equations of a
             Submersion, Michigan Mathematical Journal 13, no. 4
             (December 1966): 459–69. https://doi.org/10.1307/mmj/1028999604.
         """
@@ -388,12 +388,12 @@ def integrability_tensor_derivative(
         curvature in a submersion. The components :math:`\nabla_X (A_Y E)`
         and :math:`A_Y E` are computed at base-point :math:`P = base\_point`
         for horizontal vector fields :math:`X, Y` extending the values
-        given in argument :math:`X|_P = horizontal_vec_x`,
+        given in argument :math:`X|_P = horizontal\_vec\_x`,
         :math:`Y|_P = horizontal\_vec\_y` and a general vector field
-        :math:`E` extending :math:`E|_x =
-        tangent\_vec\_e` in a neighborhood of x with covariant derivatives
-        :math:`\nabla_X Y |_P = nabla_x_y` and
-        :math:`\nabla_X E |_P = nabla_x_e`.
+        :math:`E` extending :math:`E|_x = tangent\_vec\_e`
+        in a neighborhood of x with covariant derivatives
+        :math:`\nabla_X Y |_P = nabla_x y` and
+        :math:`\nabla_X E |_P = nabla_x e`.
 
         Parameters
         ----------

diff --git a/geomstats/geometry/full_rank_correlation_matrices.py b/geomstats/geometry/full_rank_correlation_matrices.py
@@ -86,7 +86,7 @@ def diag_action(diagonal_vec, point):
         r"""Apply a diagonal matrix on an SPD matrices by congruence.
 
         The action of :math:`D` on :math:`\Sigma` is given by :math:`D
-        \Sigma D. The diagonal matrix must be passed as a vector representing
+        \Sigma D`. The diagonal matrix must be passed as a vector representing
         its diagonal.
 
         Parameters
@@ -109,7 +109,7 @@ def from_covariance(cls, point):
         r"""Compute the correlation matrix associated to an SPD matrix.
 
         The correlation matrix associated to an SPD matrix (the covariance)
-        :math:`\Sigma` is given by :math:`D  \Sigma D` where :math:`D` is
+        :math:`\Sigma` is given by :math:`D \Sigma D` where :math:`D` is
         the inverse square-root of the diagonal of :math:`\Sigma`.
 
         Parameters