diff --git a/cartographer/mapping/internal/3d/rotation_parameterization.h b/cartographer/mapping/internal/3d/rotation_parameterization.h
index 4ce00559d2..2ac9633d7e 100644
--- a/cartographer/mapping/internal/3d/rotation_parameterization.h
+++ b/cartographer/mapping/internal/3d/rotation_parameterization.h
@@ -24,9 +24,11 @@
 namespace cartographer {
 namespace mapping {
 
-struct YawOnlyQuaternionPlus {
+// Provides operations used to create a Ceres Manifold with a 4-D ambient
+// space and a 1-D tangent space that represents a yaw rotation only.
+struct YawOnlyQuaternionOperations {
   template <typename T>
-  bool operator()(const T* x, const T* delta, T* x_plus_delta) const {
+  bool Plus(const T* x, const T* delta, T* x_plus_delta) const {
     const T clamped_delta = common::Clamp(delta[0], T(-0.5), T(0.5));
     T q_delta[4];
     q_delta[0] = ceres::sqrt(1. - clamped_delta * clamped_delta);
@@ -36,11 +38,22 @@ struct YawOnlyQuaternionPlus {
     ceres::QuaternionProduct(q_delta, x, x_plus_delta);
     return true;
   }
+  template <typename T>
+  bool Minus(const T* y, const T* x, T* y_minus_x) const {
+    T minus_x[4] = {x[0], -x[1], -x[2], -x[3]};
+    T q_delta[4];
+    ceres::QuaternionProduct(y, minus_x, q_delta);
+    y_minus_x[0] = q_delta[3];
+    return true;
+  }
 };
 
-struct ConstantYawQuaternionPlus {
+// Provides operations used to create a Ceres Manifold with a 4-D ambient
+// space and a 2-D tangent space that represents a rotation only in pitch and
+// roll, but no yaw.
+struct ConstantYawQuaternionOperations {
   template <typename T>
-  bool operator()(const T* x, const T* delta, T* x_plus_delta) const {
+  bool Plus(const T* x, const T* delta, T* x_plus_delta) const {
     const T delta_norm =
         ceres::sqrt(common::Pow2(delta[0]) + common::Pow2(delta[1]));
     const T sin_delta_over_delta =
@@ -59,8 +72,29 @@ struct ConstantYawQuaternionPlus {
     ceres::QuaternionProduct(x, q_delta, x_plus_delta);
     return true;
   }
+  template <typename T>
+  bool Minus(const T* y, const T* x, T* y_minus_x) const {
+    T minus_x[4] = {x[0], -x[1], -x[2], -x[3]};
+    T q_delta[4];
+    ceres::QuaternionProduct(minus_x, y, q_delta);
+    const T& cos_delta_norm = q_delta[0];
+    const T sin_delta_norm =
+        ceres::sqrt(common::Pow2(q_delta[1]) + common::Pow2(q_delta[2]));
+    const T delta_norm = atan2(sin_delta_norm, cos_delta_norm);
+    const T delta_over_sin_delta =
+        delta_norm < 1e-6 ? T(1.) : delta_norm / sin_delta_norm;
+    y_minus_x[0] = q_delta[1] * delta_over_sin_delta;
+    y_minus_x[1] = q_delta[2] * delta_over_sin_delta;
+    return true;
+  }
 };
 
+// Type aliases used to create manifolds.
+using YawOnlyQuaternionManifold =
+    ceres::AutoDiffManifold<YawOnlyQuaternionOperations, 4, 1>;
+using ConstantYawQuaternionManifold =
+    ceres::AutoDiffManifold<ConstantYawQuaternionOperations, 4, 2>;
+
 }  // namespace mapping
 }  // namespace cartographer
 
diff --git a/cartographer/mapping/internal/3d/rotation_parameterization_test.cc b/cartographer/mapping/internal/3d/rotation_parameterization_test.cc
new file mode 100644
index 0000000000..e2f49a81bc
--- /dev/null
+++ b/cartographer/mapping/internal/3d/rotation_parameterization_test.cc
@@ -0,0 +1,48 @@
+#include "cartographer/mapping/internal/3d/rotation_parameterization.h"
+
+#include "ceres/manifold_test_utils.h"
+#include "gtest/gtest.h"
+
+namespace cartographer::mapping {
+
+template <typename T>
+class RotationParameterizationTests : public ::testing::Test {};
+
+using TestTypes =
+    ::testing::Types<YawOnlyQuaternionManifold, ConstantYawQuaternionManifold>;
+TYPED_TEST_SUITE(RotationParameterizationTests, TestTypes);
+
+TYPED_TEST(RotationParameterizationTests, ManifoldInvariantsHold) {
+  const TypeParam manifold;
+
+  constexpr static int kNumTrials = 10;
+  constexpr static double kTolerance = 1.e-5;
+  const std::vector<double> delta_magnitutes = {0.0, 1.e-9, 1.e-3, 0.5};
+  for (int trial = 0; trial < kNumTrials; ++trial) {
+    const Eigen::VectorXd x =
+        Eigen::VectorXd::Random(manifold.AmbientSize()).normalized();
+
+    for (const double delta_magnitude : delta_magnitutes) {
+      const Eigen::VectorXd delta =
+          Eigen::VectorXd::Random(manifold.TangentSize()) * delta_magnitude;
+      EXPECT_THAT(manifold, ceres::XPlusZeroIsXAt(x, kTolerance));
+      EXPECT_THAT(manifold, ceres::XMinusXIsZeroAt(x, kTolerance));
+      EXPECT_THAT(manifold, ceres::MinusPlusIsIdentityAt(x, delta, kTolerance));
+      const Eigen::VectorXd zero_tangent =
+          Eigen::VectorXd::Zero(manifold.TangentSize());
+      EXPECT_THAT(manifold,
+                  ceres::MinusPlusIsIdentityAt(x, zero_tangent, kTolerance));
+
+      Eigen::VectorXd y(manifold.AmbientSize());
+      ASSERT_TRUE(manifold.Plus(x.data(), delta.data(), y.data()));
+      EXPECT_THAT(manifold, ceres::PlusMinusIsIdentityAt(x, x, kTolerance));
+      EXPECT_THAT(manifold, ceres::PlusMinusIsIdentityAt(x, y, kTolerance));
+      EXPECT_THAT(manifold, ceres::HasCorrectPlusJacobianAt(x, kTolerance));
+      EXPECT_THAT(manifold, ceres::HasCorrectMinusJacobianAt(x, kTolerance));
+      EXPECT_THAT(manifold,
+                  ceres::MinusPlusJacobianIsIdentityAt(x, kTolerance));
+    }
+  }
+}
+
+}  // namespace cartographer::mapping
diff --git a/cartographer/mapping/internal/3d/scan_matching/ceres_scan_matcher_3d.cc b/cartographer/mapping/internal/3d/scan_matching/ceres_scan_matcher_3d.cc
index 5aab1fd327..f2acb77f7f 100644
--- a/cartographer/mapping/internal/3d/scan_matching/ceres_scan_matcher_3d.cc
+++ b/cartographer/mapping/internal/3d/scan_matching/ceres_scan_matcher_3d.cc
@@ -31,6 +31,7 @@
 #include "cartographer/transform/rigid_transform.h"
 #include "cartographer/transform/transform.h"
 #include "ceres/ceres.h"
+#include "ceres/manifold.h"
 #include "glog/logging.h"
 
 namespace cartographer {
@@ -98,11 +99,10 @@ void CeresScanMatcher3D::Match(
   optimization::CeresPose ceres_pose(
       initial_pose_estimate, nullptr /* translation_parameterization */,
       options_.only_optimize_yaw()
-          ? std::unique_ptr<ceres::LocalParameterization>(
-                absl::make_unique<ceres::AutoDiffLocalParameterization<
-                    YawOnlyQuaternionPlus, 4, 1>>())
-          : std::unique_ptr<ceres::LocalParameterization>(
-                absl::make_unique<ceres::QuaternionParameterization>()),
+          ? std::unique_ptr<ceres::Manifold>(
+                absl::make_unique<YawOnlyQuaternionManifold>())
+          : std::unique_ptr<ceres::Manifold>(
+                absl::make_unique<ceres::QuaternionManifold>()),
       &problem);
 
   CHECK_EQ(options_.occupied_space_weight_size(),
diff --git a/cartographer/mapping/internal/imu_based_pose_extrapolator.cc b/cartographer/mapping/internal/imu_based_pose_extrapolator.cc
index ddb88ea902..dc3da7434d 100644
--- a/cartographer/mapping/internal/imu_based_pose_extrapolator.cc
+++ b/cartographer/mapping/internal/imu_based_pose_extrapolator.cc
@@ -28,6 +28,7 @@
 #include "cartographer/mapping/internal/optimization/cost_functions/spa_cost_function_3d.h"
 #include "cartographer/mapping/pose_graph_interface.h"
 #include "cartographer/transform/transform.h"
+#include "ceres/manifold.h"
 #include "glog/logging.h"
 
 namespace cartographer {
@@ -137,7 +138,7 @@ ImuBasedPoseExtrapolator::ExtrapolatePosesWithGravity(
   // we can estimate the gravity alignment of the current pose.
   optimization::CeresPose gravity_from_local(
       gravity_from_local_, nullptr,
-      absl::make_unique<ceres::QuaternionParameterization>(), &problem);
+      absl::make_unique<ceres::QuaternionManifold>(), &problem);
   // Use deque so addresses stay constant during problem formulation.
   std::deque<optimization::CeresPose> nodes;
   std::vector<common::Time> node_times;
@@ -162,13 +163,12 @@ ImuBasedPoseExtrapolator::ExtrapolatePosesWithGravity(
 
     if (is_last) {
       nodes.emplace_back(gravity_from_node, nullptr,
-                         absl::make_unique<ceres::AutoDiffLocalParameterization<
-                             ConstantYawQuaternionPlus, 4, 2>>(),
+                         absl::make_unique<ConstantYawQuaternionManifold>(),
                          &problem);
       problem.SetParameterBlockConstant(nodes.back().translation());
     } else {
       nodes.emplace_back(gravity_from_node, nullptr,
-                         absl::make_unique<ceres::QuaternionParameterization>(),
+                         absl::make_unique<ceres::QuaternionManifold>(),
                          &problem);
     }
   }
@@ -200,7 +200,7 @@ ImuBasedPoseExtrapolator::ExtrapolatePosesWithGravity(
           &imu_it_prev_prev)
           .pose;
   nodes.emplace_back(initial_estimate, nullptr,
-                     absl::make_unique<ceres::QuaternionParameterization>(),
+                     absl::make_unique<ceres::QuaternionManifold>(),
                      &problem);
   node_times.push_back(time);
 
@@ -223,7 +223,7 @@ ImuBasedPoseExtrapolator::ExtrapolatePosesWithGravity(
   std::array<double, 4> imu_calibration{{1., 0., 0., 0.}};
 
   problem.AddParameterBlock(imu_calibration.data(), 4,
-                            new ceres::QuaternionParameterization());
+                            new ceres::QuaternionManifold());
   problem.SetParameterBlockConstant(imu_calibration.data());
 
   auto imu_it = imu_data_.begin();
diff --git a/cartographer/mapping/internal/optimization/ceres_pose.cc b/cartographer/mapping/internal/optimization/ceres_pose.cc
index 807a67da68..3b28b652fa 100644
--- a/cartographer/mapping/internal/optimization/ceres_pose.cc
+++ b/cartographer/mapping/internal/optimization/ceres_pose.cc
@@ -29,14 +29,14 @@ CeresPose::Data FromPose(const transform::Rigid3d& pose) {
 
 CeresPose::CeresPose(
     const transform::Rigid3d& pose,
-    std::unique_ptr<ceres::LocalParameterization> translation_parametrization,
-    std::unique_ptr<ceres::LocalParameterization> rotation_parametrization,
+    std::unique_ptr<ceres::Manifold> translation_manifold,
+    std::unique_ptr<ceres::Manifold> rotation_manifold,
     ceres::Problem* problem)
     : data_(std::make_shared<CeresPose::Data>(FromPose(pose))) {
   problem->AddParameterBlock(data_->translation.data(), 3,
-                             translation_parametrization.release());
+                             translation_manifold.release());
   problem->AddParameterBlock(data_->rotation.data(), 4,
-                             rotation_parametrization.release());
+                             rotation_manifold.release());
 }
 
 const transform::Rigid3d CeresPose::ToRigid() const {
diff --git a/cartographer/mapping/internal/optimization/ceres_pose.h b/cartographer/mapping/internal/optimization/ceres_pose.h
index d852d80c1a..b9cbeee686 100644
--- a/cartographer/mapping/internal/optimization/ceres_pose.h
+++ b/cartographer/mapping/internal/optimization/ceres_pose.h
@@ -23,6 +23,7 @@
 #include "Eigen/Core"
 #include "cartographer/transform/rigid_transform.h"
 #include "ceres/ceres.h"
+#include "ceres/manifold.h"
 
 namespace cartographer {
 namespace mapping {
@@ -31,9 +32,9 @@ namespace optimization {
 class CeresPose {
  public:
   CeresPose(
-      const transform::Rigid3d& rigid,
-      std::unique_ptr<ceres::LocalParameterization> translation_parametrization,
-      std::unique_ptr<ceres::LocalParameterization> rotation_parametrization,
+      const transform::Rigid3d& pose,
+      std::unique_ptr<ceres::Manifold> translation_manifold,
+      std::unique_ptr<ceres::Manifold> rotation_manifold,
       ceres::Problem* problem);
 
   const transform::Rigid3d ToRigid() const;
diff --git a/cartographer/mapping/internal/optimization/optimization_problem_2d.cc b/cartographer/mapping/internal/optimization/optimization_problem_2d.cc
index 3faf2556f4..429a6a470c 100644
--- a/cartographer/mapping/internal/optimization/optimization_problem_2d.cc
+++ b/cartographer/mapping/internal/optimization/optimization_problem_2d.cc
@@ -144,8 +144,8 @@ void AddLandmarkCostFunctions(
                                          *prev_node_pose, *next_node_pose);
         C_landmarks->emplace(
             landmark_id,
-            CeresPose(starting_point, nullptr /* translation_parametrization */,
-                      absl::make_unique<ceres::QuaternionParameterization>(),
+            CeresPose(starting_point, nullptr /* translation_manifold */,
+                      absl::make_unique<ceres::QuaternionManifold>(),
                       problem));
         // Set landmark constant if it is frozen.
         if (landmark_node.second.frozen) {
diff --git a/cartographer/mapping/internal/optimization/optimization_problem_3d.cc b/cartographer/mapping/internal/optimization/optimization_problem_3d.cc
index 246707b26d..985604ea7f 100644
--- a/cartographer/mapping/internal/optimization/optimization_problem_3d.cc
+++ b/cartographer/mapping/internal/optimization/optimization_problem_3d.cc
@@ -160,8 +160,8 @@ void AddLandmarkCostFunctions(
                                          *prev_node_pose, *next_node_pose);
         C_landmarks->emplace(
             landmark_id,
-            CeresPose(starting_point, nullptr /* translation_parametrization */,
-                      absl::make_unique<ceres::QuaternionParameterization>(),
+            CeresPose(starting_point, nullptr /* translation_manifold */,
+                      absl::make_unique<ceres::QuaternionManifold>(),
                       problem));
         // Set landmark constant if it is frozen.
         if (landmark_node.second.frozen) {
@@ -274,12 +274,11 @@ void OptimizationProblem3D::Solve(
   ceres::Problem::Options problem_options;
   ceres::Problem problem(problem_options);
 
-  const auto translation_parameterization =
-      [this]() -> std::unique_ptr<ceres::LocalParameterization> {
-    return options_.fix_z_in_3d()
-               ? absl::make_unique<ceres::SubsetParameterization>(
-                     3, std::vector<int>{2})
-               : nullptr;
+  const auto translation_manifold =
+      [this]() -> std::unique_ptr<ceres::Manifold> {
+    return options_.fix_z_in_3d() ? absl::make_unique<ceres::SubsetManifold>(
+                                        3, std::vector<int>{2})
+                                  : nullptr;
   };
 
   // Set the starting point.
@@ -298,9 +297,8 @@ void OptimizationProblem3D::Solve(
       C_submaps.Insert(
           submap_id_data.id,
           CeresPose(submap_id_data.data.global_pose,
-                    translation_parameterization(),
-                    absl::make_unique<ceres::AutoDiffLocalParameterization<
-                        ConstantYawQuaternionPlus, 4, 2>>(),
+                    translation_manifold(),
+                    absl::make_unique<ConstantYawQuaternionManifold>(),
                     &problem));
       problem.SetParameterBlockConstant(
           C_submaps.at(submap_id_data.id).translation());
@@ -308,8 +306,8 @@ void OptimizationProblem3D::Solve(
       C_submaps.Insert(
           submap_id_data.id,
           CeresPose(submap_id_data.data.global_pose,
-                    translation_parameterization(),
-                    absl::make_unique<ceres::QuaternionParameterization>(),
+                    translation_manifold(),
+                    absl::make_unique<ceres::QuaternionManifold>(),
                     &problem));
     }
     if (frozen) {
@@ -324,8 +322,8 @@ void OptimizationProblem3D::Solve(
         frozen_trajectories.count(node_id_data.id.trajectory_id) != 0;
     C_nodes.Insert(
         node_id_data.id,
-        CeresPose(node_id_data.data.global_pose, translation_parameterization(),
-                  absl::make_unique<ceres::QuaternionParameterization>(),
+        CeresPose(node_id_data.data.global_pose, translation_manifold(),
+                  absl::make_unique<ceres::QuaternionManifold>(),
                   &problem));
     if (frozen) {
       problem.SetParameterBlockConstant(C_nodes.at(node_id_data.id).rotation());
@@ -550,9 +548,7 @@ void OptimizationProblem3D::Solve(
                     Eigen::AngleAxisd(
                         transform::GetYaw(fixed_frame_pose_in_map.rotation()),
                         Eigen::Vector3d::UnitZ())),
-                nullptr,
-                absl::make_unique<ceres::AutoDiffLocalParameterization<
-                    YawOnlyQuaternionPlus, 4, 1>>(),
+                nullptr, absl::make_unique<YawOnlyQuaternionManifold>(),
                 &problem));
         fixed_frame_pose_initialized = true;
       }